Falkirk 0 Celtic 3
The Bairns looked to be dealing with the Bhoys' threat fairly comfortably, but with 40 minutes on the clock Darryl Duffy was booked for a foul on restored Celtic frontman Aiden McGeady wide on the left.
And when McGeady played Mo Camara through inside the box the full-back cut the ball to the near post, where Shaun Maloney flicked home from point-blank range.
A minute later McGeady, who had forced himself back into the Parkhead first team after being left out since the first game of the season, staked his claim for a regular spot when he made space for himself 14 yards out and hit a low drive inside Matt Glennon's right hand post.
The home side made a tentative attempt to get back into the game in the 54th minute when Duffy hit a 25-yarder, but it was easily taken on the line by Artur Boruc.
The visitors were knocking the ball about pretty comfortably by this stage and made it 3-0 in the 69th minute.
It was desperately unfortunate for Falkirk goalkeeper Glennon who superbly blocked a Maloney drive, only to look up from a prostrate position to see the frightening sight of John Hartson looming - and the big frontman steered home from six yards for his 100th goal for the Hoops.
Celtic had almost broken the deadlock within 30 seconds of the start when McGeady robbed Neil Scally before cutting low across the face of goal. Home defender Andrew Lawrie, however, came to his side's rescue by sticking out a boot to divert the ball behind for a fruitless corner.
The first period failed to produce much goalmouth incident until the dramatic 41st and 42nd minute, and although Falkirk never gave up the ghost the game was well and truly over by the time Hartson's landmark goal was scored.