The Irishman doubled his tally with a cute flick 11 minutes from the end after latching on to a long-range Marc Crosas free-kick.
Vitor Lima had a clutch of good chances to open Falkirk's account, but failed to hit the mark with all three efforts.
The win kept Falkirk rooted to the bottom of the table, three points adrift of Hamilton, while Celtic continue to hope for a Rangers slip-up to keep their title chances alive.
Keane's strike in the fourth minute was deflected out by Falkirk keeper Robert Olejnik for a corner, then Enoch Showumni headed over the Celtic crossbar from eight yards out at the other end.
The London-born Nigerian put Celtic keeper Artur Boruc under pressure on the counter-attack in the 14th minute and forced a corner, but the hosts failed to capitalise on the advantage.
Falkirk enjoyed a good spell of pressure as the match settled down, putting Celtic on the back foot and Lima came close in the 20th minute with a route-one strike from the edge of the D that Boruc managed to smother.
The Portuguese midfielder followed up his effort with a volley that sailed over the bar after he broke free and latched on to a searching corner.
As the half hour mark approached, Keane broke free to collect a threaded ball from Morten Rasmussen, firing his late shot of the legs of Olejnik.
Boruc had to be on full alert to collect a Darren Barr header, then Olejnik did well to parry out a powerful left-footed Keane strike from 18 yards.
The hosts continued to look dangerous on the counter-attack, and Scott Arfield was unlucky not to reach a Ryan Flynn cross.
Keane broke the deadlock with his fourth goal of his loan spell from six yards out in the 34th minute when Rasmussen split the Falkirk defence and the Republic of Ireland captain slotted home.
As the half-time interval approached, Boruc kept out a Flynn overhead kick as the hosts pushed for a leveller.
On the stroke of half-time, Showunmi passed up a chance to restore parity after some good work by Arfield, sclaffing his shot wide of Boruc's left-hand post.
The tempo of the first half dropped after the restart, although a Crosas corner threatened the Falkirk goal, but Olejnik made the save.
Barr threatened at the other end with a goal-line cross that Darren O'Dea was quick to clear.
As the hour mark approached, Aiden McGeady blasted in a powerful shot that Tam Scobbie blocked, then a Georgios Samaras effort inched wide.
Falkirk continued to press for an equaliser but Showumni was lacking support from his fellow Bairns up front.
In the 68th minute, Barr crossed for Flynn to head towards the Celtic goal, but it sailed over.
Lima should have levelled the scores with 20 minutes to play when he turned near the penalty spot and lofted his shot over the crossbar.
Boruc kept out a long-range drive from Arfield with 12 minutes to play, then Keane expertly executed his second goal of the day to double Celtic's tally.
The on-loan Spurs man beat the offside trap to collect a long-range Crosas free-kick and skip past the drowsy Falkirk defence and flick a cute shot past Olejnik.
Arfield came close with another long-range shot as the final minutes ticked out but Boruc was enjoying a comfortable day in the Celtic goal and held well.
Live text and stats
Sunday, 7 March 2010
06 McLean yellow card
04 Twaddle yellow card
23 Lima (Moutinho 80)
09 Samaras (McGowan 90)
19 Rasmussen (Wilson 90)
Ref: WinterAtt: 6,792
Attempts on target
Attempts off target
90:00+1:22 The final whistle goes and the game is over.
89:28 Substitution Mark Wilson replaces Morten Rasmussen.
89:28 Substitution Paul McGowan on for Georgios Samaras.
89:05 Pedro Moutinho takes a shot. Blocked by Josh Thompson.
86:42 The assistant referee signals for offside against Robbie Keane. Direct free kick taken by Robert Olejnik.
84:38 Andreas Hinkel gives away a free kick for an unfair challenge on Ryan Flynn. Brian McLean takes the free kick.
83:55 Scott Arfield takes a shot. Save by Artur Boruc.
80:24 Edson Braafheid delivers the ball from the free kick left-footed from right wing, Robert Olejnik makes a save.
80:24 Booking Booking for Marc Twaddle.
80:15 Marc Twaddle challenges Morten Rasmussen unfairly and gives away a free kick.
79:42 Substitution Pedro Moutinho is brought on as a substitute for Vitor Lima.
78:36 Marc Crosas provided the assist for the goal.
78:36 GOAL - Robbie Keane:Falkirk 0 - 2 Celtic Robbie Keane finds the net with a goal from inside the penalty area to the bottom right corner of the goal. Falkirk 0-2 Celtic.
78:26 Unfair challenge on Landry N'Guemo by Scott Arfield results in a free kick. Free kick taken by Marc Crosas.
77:59 Artur Boruc takes the direct free kick.
77:59 Booking Brian McLean booked for unsporting behaviour.
77:14 Inswinging corner taken from the left by-line by Ryan Flynn, Artur Boruc fouled by Brian McLean, the ref awards a free kick.
76:56 Scott Arfield takes a shot. Save made by Artur Boruc.
76:00 Foul by Scott Arfield on Aiden McGeady, free kick awarded. Landry N'Guemo takes the direct free kick.
74:59 Outswinging corner taken by Edson Braafheid, clearance made by Enoch Showunmi.
72:07 Vitor Lima takes a shot from just inside the penalty box which clears the crossbar.
71:17 Unfair challenge on Darren Barr by Georgios Samaras results in a free kick. Robert Olejnik takes the direct free kick.
70:46 Thomas Scobbie gives away a free kick for an unfair challenge on Robbie Keane. Andreas Hinkel restarts play with the free
69:47 Aiden McGeady gives away a free kick for an unfair challenge on Ryan Flynn. Ryan Flynn delivers the ball from the free kick
left-footed from left wing, Georgios Samaras manages to make a clearance.
67:45 Darren Barr crosses the ball, Close range header by Ryan Flynn goes over the crossbar.
66:44 Free kick awarded for an unfair challenge on Ryan Flynn by Marc Crosas. Thomas Scobbie restarts play with the free kick.
66:16 Free kick awarded for an unfair challenge on Andreas Hinkel by Scott Arfield. Marc Crosas takes the free kick.
64:43 Corner taken by Colin Healy from the left by-line to the near post, save made by Artur Boruc.
63:51 Corner taken by Colin Healy from the left by-line to the near post, Landry N'Guemo manages to make a clearance.
59:37 The assistant referee signals for offside against Enoch Showunmi. Artur Boruc takes the indirect free kick.
58:47 Aiden McGeady takes a shot. Thomas Scobbie gets a block in. Georgios Samaras has an effort at goal from outside the area which
goes wide right of the goal.
55:50 Free kick awarded for a foul by Darren O'Dea on Colin Healy. Free kick crossed by Burton O'Brien, Georgios Samaras manages
to make a clearance.
54:59 Ryan Flynn produces a cross, Josh Thompson manages to make a clearance.
53:49 Outswinging corner taken by Colin Healy from the right by-line, Josh Thompson makes a clearance.
51:56 Inswinging corner taken right-footed by Marc Crosas, Robert Olejnik makes a save.
50:29 Free kick awarded for an unfair challenge on Brian McLean by Morten Rasmussen. Free kick taken by Robert Olejnik.
50:00 Morten Rasmussen challenges Thomas Scobbie unfairly and gives away a free kick. Robert Olejnik restarts play with the free
45:01 The referee starts the second half.
45:00+0:03 The ref blows to signal half-time.
44:18 Effort from deep inside the area by Enoch Showunmi misses to the right of the target.
41:16 Foul by Scott Arfield on Morten Rasmussen, free kick awarded. Free kick taken by Landry N'Guemo.
38:13 Free kick awarded for a foul by Colin Healy on Marc Crosas. Direct free kick taken by Artur Boruc.
37:13 Corner taken by Colin Healy from the right by-line, Ryan Flynn has an effort at goal from 6 yards. Artur Boruc makes a fantastic
save. Short corner worked by Colin Healy.
36:42 Foul by Landry N'Guemo on Colin Healy, free kick awarded. Burton O'Brien takes the free kick.
35:37 The referee penalises Morten Rasmussen for handball. Indirect free kick taken by Colin Healy.
33:45 Assist on the goal came from Morten Rasmussen.
33:45 GOAL - Robbie Keane:Falkirk 0 - 1 Celtic Robbie Keane grabs a goal from close range low into the middle of the goal. Falkirk 0-1 Celtic.
32:20 A cross is delivered by Ryan Flynn.
30:42 Shot by Robbie Keane. Robert Olejnik makes a save. Inswinging corner taken by Marc Crosas, clearance made by Marc Twaddle.
Marc Crosas takes a outswinging corner from the right by-line to the near post, Effort on goal by Josh Thompson from 12 yards.
Darren Barr gets a block in.
29:40 Darren Barr takes a shot. Save made by Artur Boruc.
29:05 Corner taken by Colin Healy from the left by-line to the near post, clearance made by Landry N'Guemo.
27:44 Edson Braafheid takes the inswinging corner, save by Robert Olejnik.
25:49 Centre by Aiden McGeady, Brian McLean makes a clearance.
25:09 Robbie Keane takes a shot. Robert Olejnik makes a save. Edson Braafheid takes a short corner.
21:04 Outswinging corner taken right-footed by Colin Healy played to the near post, Shot on goal by Vitor Lima from inside the box
goes harmlessly over the target.
19:39 Shot from just outside the box by Vitor Lima misses to the left of the goal.
18:46 Georgios Samaras gives away a free kick for an unfair challenge on Scott Arfield. Brian McLean takes the free kick.
17:08 Foul by Georgios Samaras on Darren Barr, free kick awarded. Direct free kick taken by Darren Barr.
14:30 Corner taken by Colin Healy from the right by-line, clearance by Georgios Samaras.
7:45 Unfair challenge on Darren O'Dea by Enoch Showunmi results in a free kick. Direct free kick taken by Josh Thompson.
6:41 Georgios Samaras takes a shot. Thomas Scobbie gets a block in. Corner taken by Edson Braafheid, clearance made by Marc Twaddle.
5:15 Scott Arfield delivers the ball, Enoch Showunmi has a headed effort from close in and clears the bar.
3:40 Robbie Keane produces a drilled right-footed shot from 20 yards. Burton O'Brien gets a block in. Edson Braafheid takes the
inswinging corner, clearance by Darren Barr.
0:20 Foul by Colin Healy on Landry N'Guemo, free kick awarded. Artur Boruc takes the free kick.
0:00 The match begins.
Live text and data provided by The Press Association.
Source: BBC Sport