The winner was fittingly crafted by their best performers, as Craig Sibbald headed in Blair Alston's cross.
Scott Allan excelled for Hibs and hit the crossbar in the second half.
Fraser Fyvie had struck the post before then and other chances were scorned, while Falkirk decisively took the one that came their way.
Peter Houston's side will return to Hampden on 30 May to face the winner of Sunday's semi-final between Inverness Caledonian and Celtic.
The intent of Hibs was immediately apparent, as Jason Cummings gathered possession just inside the Falkirk half, nutmegging an opponent, and surged into the area but then delivered the cut-back behind his strike partner Farid El Alagui.
The attack was sharp and assertive, and the entire moment turned out to be emblematic of the game itself.
Falkirk were lively in spells during the first half, mostly when Sibbald or Alston were in possession, but Hibs grew into dominance.
But good fortune seemed to take allegiance with Falkirk. When Dylan McGeouch delivered a cross, Falkirk defender David McCracken stretched to meet it ahead of El Alagui and sent the ball spinning towards his own goal, and Jamie MacDonald had to react quickly to block it. The rebound bounced off the grounded El Alagui, and MacDonald instinctively reacted to stop the ball again.