The home side passed up several decent chances before Keane grabbed his second in the 79th minute to take the Parkhead club back to within 10 points of SPL leaders Rangers.
The Ibrox side also have a game in hand and, barring an unlikely collapse, are poised to romp to their second successive title.
However, after drawing twice against the Bairns earlier in the season, at least Tony Mowbray survived another potential embarrassment - and he can thank Keane, who showed why he is reportedly being paid £65,000 per week by the Parkhead club for his loan spell until the end of the season.
Before kick-off there was a perfectly-observed minute's silence as a tribute to former Falkirk player Jack McAndrew, who died last week.
In a lukewarm start to the game, the Bairns survived an early Celtic corner before Falkirk's giant striker Enoch Showunmi headed a Scott Arfield cross over the bar at the other end.
And in the 20th minute midfielder Vitor Lima fired wide from the edge of the box after decent build-up play again involving Arfield, who was looking lively.
The Hoops had on Friday refuted reports that they had voiced concerns to the SPL about the state of Falkirk's pitch. The playing surface was, though, in poor condition but Celtic fashioned a good chance when Keane beat the home side's offside trap to race on to Andreas Hinkel's through-ball.
With the away fans behind the goal ready to celebrate, Olejnik made a good block from the Irishman to concede a corner, which Falkirk survived.
The pressure was building and in the 31st minute Olejnik made a great save from Keane after the on-loan Hoops striker had unleashed a ferocious shot from 16 yards.
However, three minutes later Keane showed his class when Rasmussen threaded him in, this time the Republic of Ireland striker confidently knocking the ball low past Olejnik.
Falkirk had threatened sporadically beforehand but moments later Celtic keeper Artur Boruc had to stretch on the line to claw away Ryan Flynn's close-range effort and prevent a certain equaliser.
However, seconds before the break the Celtic defence appeared frail when Arfield drove into the box but, when the ball broke fortuitously for Showunmi, he scuffed his shot with only Boruc to beat.
It was a scrappy start to the second half with Celtic taking time to get back into their stride.
In the 55th minute, as Falkirk pressed, Darren O'Dea made a timely clearance after Flynn beat Hinkel on the left to stand the ball up in the six-yard box.
On the hour mark, Georgios Samaras flashed a shot from the edge of the box wide of the target and, seconds later, Showunmi threatened to burst clear into the visitors' box but the Bairns' lone striker did not have the change of gear and was again crowded out.
Showunmi missed another decent chance when he headed Darren Barr's cross over the crossbar from six yards out and then, in the 72nd minute, the Scotland defender set up Lima who drove over from 14 yards.
Boruc then produced a fine save at his right-hand post from Arfield's 20-yard volley after the Falkirk midfielder had been set up by Showunmi.
But the home side were made to pay for their profligacy in the 79th minute when a quick Marc Crosas free-kick sent Keane racing past the Bairns defence and he knocked the ball past Olejnik to secure the points.
Source: Team Talk
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