Hoops midfielder Aiden McGeady had a first-half penalty saved by Robert Olejnik, before Gary Caldwell headed the visitors ahead just after the break.
Substitute Scott McDonald, on moments earlier for Paddy McCourt, then headed the equaliser in the 72nd minute before grabbing another five minutes later.
But there was still time for Bairns substitute Mark Stewart to equalise as all six goals came in the second period.
Falkirk head coach Eddie May praised his injury hit squad.
"Football's a funny thing - you can put out your strongest team and get beaten," May told Sky Sports.
"We deserved a point. Should we have won it? Well, Celtic created a number of chances so I think a draw was about right."
Falkirk are still without a win this season and remain bottom but May was heartened by his team's fighting qualities.
He added: "It's important to show the home crowd that we are willing to do the things needed to stay in the league."
Celtic boss Tony Mowbray admitted the Europa League trip to Hamburg might have resulted in a lacklustre performance from his players.
He told Sky Sports: "I'm trying to find excuses. We had a high-energy, high-emotion game against Hamburg on Thursday.
"I think we know as a team we have to defend better, and it's very disappointing to concede three goals."