The Jambos progressed to the final four with a 1-0 win at Easter Road thanks to a 34th minute Ryan Stevenson goal before James McPake was sent off for Hibs in the dying minutes.
Hearts boss Gary Locke admitted he was extremely proud of his young group of players for the way they went about their business in the vital derby win.
"I think it is considering the age of the squad. It's a fantastic result for us,” he said.
"It was a great performance, apart from the first 15 minutes, and we've got to start taking that type of performance into the league games.
"If we put in performances like that hopefully we can turn our season around."
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