Indeed the home side were one down at the interval after Darryl Duffy had put the visitors ahead from the penalty spot.
Celtic drew level soon after half-time from a John Hartson header. Two superb strikes by Alan Thompson in the last 15 minutes ensured all three points remained at Parkhead but the result rather flattered a stuttering performance from the Bhoys.
The home team looked slow and sluggish throughout the first half and if Thompson, who had been taking some stick from the home fans had not struck late on, then Falkirk could have left with a deserved share of the points.
Celtic pushed forwards from the kick-off and Shunsuke Nakamura, who had passed a late fitness test, was just too strong with his cross for Hartson to reach.
When Daniel McBreen's cross was deflected to Steve Thomson the youngster snatched hurriedly at his shot with his attempted volley going well wide.
Celtic created the first real opening after 10 minutes when Mo Camara's cross broke off Hartson to Maciej Zurawski, whose first-time shot was turned behind for a corner by keeper Matt Glennon at full stretch.
Thompson then had a poor first touch when Hartson's header found him in space in the box, but the ball was eventually cleared for a corner.
A Thompson cross was cleared to Paul Telfer on the edge of the box and the wing-back flicked the ball up before volleying a shot which was headed for the top corner until Glennon managed to get a hand to it and touch it over.
It looked a matter of time before Celtic made the breakthrough but strangely the home team lost its way and Falkirk came more into the match.
And after a swift counter attack Artur Boruc in the Celtic goal had to throw himself to the ground to gather a long-range shot from Thomson.
Craig Ireland nearly conceded a soft opener to Celtic when he struck his pass back far too hard, but Glennon did well to side-step the onrushing Zurawski before clearing.
Falkirk took the lead from the penalty spot after Neil Lennon lost possession on the halfway line. McBreen curled in an exquisite pass to Duffy who was brought down by Camara from behind on the six-yard line.
The Falkirk striker wrongfooted Boruc before placing his spot-kick into the corner of the net.
Thompson's crossfield pass found Stilian Petrov as Celtic tried to respond but he shot over.
Tiago had been superb in the first half in keeping Hartson under control, but the Portuguese centre-back lost his man as Celtic equalised shortly after the break.
Nakamura managed to do what had eluded Camara throughout the first half and sent in a dangerous cross which allowed Harton to rise unchallenged at the back post before heading home.
Minutes before, the Japanese midfield player had sent in another superb cross but the ineffective Zurawski managed to get in Petrov's way and was substituted by Shaun Maloney shortly after.
After weathering the storm, Falkirk passed up a great chance to restore their lead when Duffy failed to control Craig McPherson's clever pass and shot weakly wide.
Thompson was booked after a bit of petulance, but shortly afterwards the Englishman showed his good side when he fired in an unstoppable volley from 20 yards from Hartson's lay-off on the edge of the box.
In the last minute Thompson wrapped up the points when he curled his free-kick into the top corner after Petrov had been fouled by Tiago at the edge of the area.