The Jambos managed to convert a late penalty in a dramatic end to the Hampden Park clash which has allowed them to set up an Edinburgh derby final with Hibernian for the first time since 1896.
Rudi Skacel opened the scoring for Hearts early in the second half when he ran onto a ball from Craig Beattie to slot home and it appeared as though Paulo Sergio's men would be heading to the decider with a 1-0 win until Gary Hooper popped up with an 86th minute header from Charlie Mulgrew's cross to give the league champions some hope.
But disaster struck for the Hoops in the final minute of regular time when Hearts skipper Marius Zaliukas' attempt hit the arm of Joe Ledley, prompting referee Euan Norris to point to the spot with Beattie stepping up and slotting home to give the Edinburgh outfit the win.
It was a result that denied Neil Lennon's Celtic side a chance of a domestic double after losing the League Cup to Kilmarnock last month but gives Hearts the opportunity to add to their seven Scottish Cups with their most recent triumph being in 2006.
The final against Hibs, who beat Aberdeen 2-1 yesterday, sets up a fourth Edinburgh derby of the season with Hearts taking in a three-win advantage on the back of an undefeated run of 10 (which includes seven victories) against their bitter enemies.