The Bairns won 6-5 in the shootout after the sides couldn't be separated over the 90 minutes, with Robbie Mutch saving the decisive spot kick from Jack Aitchison - his second crucial penalty save of the night.
Falkirk, who had eight changes from the weekend, opened the scoring through Louis Longridge on 10 minutes as he reacted quickest to fire the ball home from Moore's cross, which had been spilled by Mullen under pressure from Johnstone.
Celtic though levelled immediately through Robertson, and it was a nice worked move which engineered space for the midfielder to curl the ball into the bottom corner from the edge of the box, with Ewan Henderson the architect.
Celtic then won a penalty after Armstrong Oko-Flex was brought down in the box by Longridge. However, Mutch, in for Cammy Bell, made a fine save, diving to his left to deny Robertson a second.
The visitors were looking dangerous particularly through Oko-Flex down the left-hand side, but it was down the opposite wing that they created another gilt-edge opportunity as Aitchison turned the ball wide from Ralston's delivery.
That was the way it remained at the break but Falkirk had a great double chance just before the hour mark to restore their advantage.
Firstly Morgaro Gomis was inches away from connecting with MacLean's cross, then Longridge picked up the loose ball, cut inside and sent an effort fizzing wide of the goal.
Celtic almost got their noses in front but Buchanan made an important block a yard or two from his goalline to deny Robertson's goal-bound effort.
The Hoops should have won this game with minutes remaining when substitute Burt spurned a great chance one on one with Mutch
And they were made to rue that as Falkirk came out on top in spotkicks thanks to the deputy shot-stopper.