Hoops midfielder Aiden McGeady had a first-half penalty saved by Bairns keeper Robert Olejnik before Gary Caldwell headed the visitors ahead just after the break.
That was the start of a goal avalanche which left supporters of both sides breathless by the final whistle.
Scott Arfield levelled from the spot after Caldwell had tripped Ryan Flynn, before Pedro Moutinho, re-signed by Falkirk following a spell at CS Maritimo, volleyed the home side ahead.
Parkhead substitute Scott McDonald, a 67th minute replacement for Paddy McCourt, headed the equaliser five minutes later before grabbing a second and at that point, it looked all over for Eddie May's side.
However, with seven minutes remaining, Stewart, on for Alex MacDonald, ran on to a clever pass by Flynn to slam the ball past Celtic keeper Lukasz Zaluska.
Mowbray's side went two points clear of Rangers at the top of the table but the Ibrox men have a game in hand.
The Celtic boss put this latest blip down to an exacting trip to Hamburg on Thursday night.
"It was a tired performance from us," he said. "It was probably as poor as we've been this season and the team didn't play well in the first half.
"It was the same defence which kept a clean sheet in Hamburg but mentally it was a tough day for the team.
"Generally, we have had mentally tired performances after games in Europe and I think that was one today.
"Yet, if we want to take a positive, the team tried to rally themselves from a losing position and showed great endeavour to get back into a winning position.
"They could have felt sorry for themselves, they had a tough week, getting back from Europe in the early hours (of Friday).
"Obviously we are disappointed to drop points but we have to move on."
On McDonald, who was dropped from the side which drew in Hamburg, Mowbray said: "There is competition for places.
"I'm delighted that he scored a couple of goals.
"From my perspective you'd like to be in the position where you don't leave Scott on the bench because he scores every single week.
"But he has had a few weeks where he has had some chances and the keeper has made good saves but hopefully those goals will help his confidence."
Falkirk remain bottom of the table with five points from 11 games but manager May, who has been under pressure in recent weeks, was pleased with his side's performance as much as the result.
He said: "We have to ask every player to play to their maximum and if we can do that we have a chance of staying in the league.
"I am just pleased when guys play to their potential and if they do that then they have the chance to make a good living from the game.
"People have said I have brought too many young guys into the club but the reason I have done that is that they are all I can afford.
"But the ones that come in need to have the character and ability to play week in, week out.
"Sometimes it happens for them, sometimes it doesn't."