After a goalless first half, Gary Caldwell put the visitors in front with a header after Robert Olejnik had kept out Aiden McGeady's penalty.
Falkirk were then awarded a penalty of their own which was converted by Scott Arfield, before Pedro Moutinho put the home side in front.
Celtic were back ahead though, after second-half substitute Scott McDonald scored two goals inside of ten minutes, but with the game reaching its conclusion, Mark Stewart struck to level the scores.
The result leaves Celtic just two points ahead of Rangers in the SPL title race, with the blue side of Glasgow having a game in hand and a chance to leapfrog their bitter rivals.
Falkirk, despite their hard-earned point, remain at the foot of the table with just five points from 11 games, four points adrift of Hamilton Academicals, Kilmarnock and St Johnstone.
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Henry caps EPL swansong on perfect day for Arsenal
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Should 'Vanishing Spray' be introduced globally?
The spray, which has been designed to stop defensive walls from encroaching closer than 10 yards to the ball at free-kicks, is set to be discussed at next month's IFAB meeting
