The striker nodded in Blair Alston’s cross from the right from six yards out to snatch victory and leave Hibs ruing their missed opportunities having dominated the game at Hampden Park.
In a scrappy first half, Falkirk had goalkeeper Jamie MacDonald to thank early on for preventing an own goal by blocking a mishit clearance, before saving Farid El Alagui’s close-range header from the rebound.
Hibs had the bulk of the possession with Dylan McGeouch playing a neat one-two but dragging his shot wide from the edge of the box, while El Alagui’s first-time shot from a tight angle flew into the side-netting.
MacDonald again came to the rescue by getting his fingertips to Fraser Fyvie’s long-range drive which hit the post.
Falkirk’s best chance of the half fell to David McCracken, who headed wide at the near post following a corner.
The Bairns improved at the start of the second period and went close when Alston’s long-range effort was pushed around the post.
Hibs should have scored, though, when McGeouch mishit the ball from close range before crossing for Lewis Stevenson, who blazed his volley over.
Alan Stubbs’ side continued to up the ante and hit the crossbar a few minutes later through Scott Allan’s fierce effort from inside the box, while Stevenson’s deflected effort nestled into the side-netting.
And Falkirk made them pay for those missed chances on 75 minutes through Sibbald’s close-range header.
MacDonald then produced a fine save to keep out El Alagui’s low shot in the dying minutes as Peter Houston’s side held on for victory, the third time they have beaten Hibs 1-0 this season.
They now await the winners of Sunday’s second semi-final between Inverness Caledonian Thistle and Celtic.