The Bairns took a first-half lead through Stuart Murdoch, but were made to pay for a number of missed chances when Alan Cook levelled with 22 minutes remaining.
Skipper Darren Dods went close to giving Falkirk an early lead but his header was kept out by Kenny Arthur, who made a good block midway through the first-half from Lyle Taylor.
The breakthrough came 10 minutes before the break when Sean Higgins was bundled over in the box and Murdoch made no mistake from the penalty spot.
They could have doubled their lead moments later when Taylor was sent clear but Cameron MacDonald slid in with a goal-saving tackle to keep it 1-0 at half-time.
United came out stronger after the break and Kieran Duffie saved Falkirk with a brave block as Ryan Donnelly looked to convert from only four yards out.
Michael McGovern made a fine save from Nathan Blockley but he was involved in Airdrie's leveller when he chased a seemingly lost cause and hooked across for Cook to bundle the ball home.
Falkirk had one final chance to take the points in the closing seconds but the lively Taylor fired narrowly wide, leaving them with only one win and six points from their opening six games.