Aiden McGeady helped himself to a hat-trick - his first for the club - after Scott McDonald opened the scoring with an early penalty and Falkirk's cause was not helped by a sending-off for defender Darren Barr after 63 minutes after he picked up two yellow cards.
Celtic gave a debut to 20-year-old Ryan Conroy as the home side started strongly with Jan Vennegoor of Hesselink just failing to get on the end of a McGeady cross.
It did not take them long to make the breakthrough. A clumsy tackle by Kenny Milne on McDonald inside the area brought a penalty after nine minutes and McDonald sent Tim Krul the wrong way with the spot-kick.
McDonald missed an easy chance shortly afterwards before Gary Caldwell thundered in a 25-yard shot which Krul did well to touch past for a corner.
But Falkirk almost equalised after 17 minutes when Scott Arfield had a chance after good work from Pedro Moutinho, but Caldwell made a great blocking challenge.
There was another let-off for Celtic shortly afterwards when a flick from Gerard Aafjes appeared to strike Conroy's arm on the lin,e but referee Dougie McDonald waved play on.
Falkirk had worked their way back into the game and Moutinho tried his luck from distance but it was always rising and Mark Brown was untroubled. Michael Higdon then linked well with Graham Barrett before directing his shot straight at Brown.
But it was Celtic who extended their lead four minutes before half-time. McGeady took advantage of some hesitant play at the back by Falkirk to drill home a 15-yard shot off the inside of the post.
It was harsh on Falkirk who had recovered from a nervy start to trouble Celtic on more than one occasion.
Celtic had a strong penalty claim turned down themselves two minutes into the second half when Milne looked to handle on the line to block a Vennegoor of Hesselink effort.
The roof fell in on Falkirk shortly afterwards. Barr was red-carded after 63 minutes when he picked up his second booking for pulling back McGeady.
Paul Hartley then struck the underside of the crossbar with a terrific effort before McGeady grabbed his second.
He cut inside two defenders before curling an unstoppable shot past Krul to make it 3-0.
McGeady completed his hat-trick eight minutes from time when he cashed in on good work from substitute Chris Killen to produce a cool finish.