St Mirren defeated their final opponents 2-0 at St Mirren Park to leapfrog John McGlynn's side in the table.
Paul McGowan blazed home from the penalty spot after three minutes and Graham Carey's wonder goal extended their lead on the stroke of half-time.
The Tynecastle side rarely threatened during a disappointing showing - their fourth successive defeat in the league.
"It'll be a different ball game at Hampden, a cup final and a one-off," said McGlynn.
"It's a different situation, different place, different crowd, bigger atmosphere, a lot more involved in the game.
"I'd have much rather have won the game and taken it into that game, but the defeat doesn't decide the League Cup final.
"It was a disappointing night for us. The first goal was a killer, to lose such a cheap goal so early."