Lyle Taylor scored a second-half hat-trick for the visitors, adding to Jay Fulton opener - with Ryan Donnelly briefly levelling the game.
Fulton broke the deadlock on 43 minutes when he collected a pass from Keiran Duffy and his shot on the turn past the keeper.
Airdrie rallied when a free-kick was knocked down to Donnelly (56) and he fired home from close range.
But parity lasted for just 11 minutes before the Bairns struck again - this time Taylor drilled a low shot into the corner when he latched on the a defence splitting pass from Euen Grant.
United suffered a major setback when Nathan Blockley was issued with a second yellow card after a needless foul on Fulton 11 minutes from time.
And Falkirk added to Airdrie's misery within 60 seconds as Taylor pounced from the resulting free-kick by Grant before completing his treble in the dying seconds.