A 90th minute strike from captain David McCracken was enough to secure all three points for Falkirk in a match that looked like heading for a draw.
The Bairns had passed up several good goalscoring opportunities in either half before the defender drilled the ball home from the edge of the box with only seconds left to play.
The win moves Peter Houston’s side up to fifth in the Championship table, just two points away from Hibs.
The Edinburgh side created several chances of their own and will be disappointed not to have left Westfield with at least a draw, but they lacked a cutting edge in the final third.
In-form striker Dominique Malonga was suspended, and Jason Cummings was substituted at half-time by Hibs boss Alan Stubbs after struggling to make an impact.
Falkirk’s best chance of a scrappy first half fell to Rory Loy with half an hour played. The striker raced clear and was about to pull the trigger when he was denied by a superb late challenge from Paul Hanlon.
Sam Stanton almost put the away side in front when his shot was deflected just wide of the post shortly before half time.
Loy had another opportunity to make the breakthrough on 51 minutes when the ball broke to him from a Sibbald free kick, but he could only lash it over the bar from 12 yards.
Five minutes later Hibs went close again when Danny Handling forced a smart save from Jamie MacDonald. The danger wasn’t over for the Bairns as the ball then fell to Paul Heffernan, but his shot was well blocked on the line by Luke Leahy.
With the clock counting down it looked like both teams would have to settle for a point - before captain fantastic David McCracken’s late intervention.
Falkirk: MacDonald, Duffie, McCracken, Vaulks, Taiwo (Cooper 68), Alston (Shepherd 80), Sibbald, P. Grant, Leahy, Biabi (Smith 65), Loy
Hibernian: Oxley, Gray, Hanlon, Fontaine, Robertson, Stanton, Heffernan (Harris 80), Stevenson, Handling, S. Allan, Cummings (L. Allan 45)
Referee: Stephen Finnie