This came after substitute Craig Beattie's last-minute penalty gave the Jambos a shock 2-1 win over his former side Celtic at Hampden.
A minute after the break Rudi Skacel raced on to a pass from Beattie, who had replaced Scott Robinson at the interval, to fire Hearts into the lead.
Four minutes from time Celtic striker Gary Hooper headed in a Charlie Mulgrew cross, but in the final minute referee Euan Norris dramatically awarded Hearts a penalty, when Marius Zaliukas' shot hit the arm of Joe Ledley, and Beattie scored from the spot.
After the game Beattie said Hearts deserved their place in the final.
"It was fantastic, a great moment for myself and we deserved it," Beattie told Sky Sports 1. "We came with a gameplan to frustrate Celtic and I think we did that.
"To come back after they scored to equalise, I'm delighted.
"I was thinking please dive out of the way quick enough when I stepped up to take it. I don't normally take penalties but I've practised a couple during the week."
The striker is now looking forward to a sell-out crowd in next month's showdown after ending Celtic's hopes of retaining the cup and claiming the double.
"The place will be rocking," he said. "It will be sold out and it's going to be a great day to compete for the Scottish Cup.
"It's going to be a fantastic day. Favourites go out of the window."
Hearts manager Paulo Sergio admitted he took a risk when he replaced Scott Robinson with Beattie at half time.
"In the first half we chose a strategy and I believed in it over the 90 minutes," Sergio said.
"I knew we were going to defend more in the first half but even with this strategy I thought we would have more ball in the first half and I didn't want to wait.
"I took the risk at half-time as I had to change things as we were not threatening the Celtic defence and it worked very well for us."
Celtic first team coach Alan Thompson called into question the refereeing decisions.
The Hoops had a penalty shout of their own turned down just after Beattie's goal when the ball appeared to come off the arm of Andy Webster.
"We are very, very disappointed to say the least, more for the players and the fans," Thompson told Sky Sports News.
"It's inconsistency from the referee. If one's a penalty the other one's a penalty.
"Initially we thought the Hearts one wasn't a penalty. The ball came up to Joe Ledley's hand and it was so close to his hand he couldn't get out of the way.
"If that was a penalty we thought the Webster one should have been a penalty.
"There's an inconsistency with how the referee sees it. Yet again we think we've been done in a cup final from a big decision. In the League Cup final we were denied a stonewall penalty.
"On the whole we're disappointed and thought we deserved more.
"We obviously wanted to go on and finish the season on a high but that won't take away what we've done in the league."