Craig Gordon was a surprise inclusion over a year after his his double leg break, and Lawrence Shankland was left out because of a virus.
The Spartans, who beat Arbroath in the last round, passed the ball around, forcing the home side back, but they were unable to pose a significant threat.
That was until Kenneth Vargas opened the scoring in the second phase after a penalty after 11 minutes. Alex Cochrane picked up a loose ball at the edge of the penalty box and he sent the ball left to Jorge Grant. The midfielder chipped the ball with his right foot into the danger area and the Costa Rica-born Vargas rose to nod home.
Spartans were only able to launch spasmodic attacks but the Hearts rearguard mopped up until Callum Booth let fly with his left food from just outside the box and Gordon was forced to dive full length to palm the ball over the bar. Time 21 minutes.
Two minutes later Alan Forrest latched on to the ball just outside the Spartais box and fired for goal with his right food. Spartans goalkeeper Blair Carswell got down quickly to save.
Hearts remained in control, patiently passing the ball around, and the Jam Tarts almost broke through two minutes from the break.
Cochrane sent the ball into the box and Vargas let fly only to see the Spartans goalkeeper block from close range.
The scoreline stayed the same untyil the break but Carswell threw himself full-length to deny Vargas a second goal as Hearts maintained their hold on the game.
Kent had a free header minutes later but nodded wide of Carswell’s right-hand post and Craig Halkett nodded goalwards and the ball was scrambled away.
Hearts introduced new signing Dexter Lembikisa and teenage talent Wilson and Cochrane had an opportunity from range but Carswell was equal to the task.
From nowhere, after 65 minutes, Spartans were back in the game from a Booth corner. He floated the ball to the edge of the box and James Craigen fired home with his instep, a goal which did not look likely but turned the game. Gordon did not move.
So Hearts, who were in cruise control, were back under severe pressure and Lembikisa was a threat on the right-hand side, delivering a rtesting cross which his team-mates failed to capitalise on.
However, Grant was inches away from a free-kick just outside the box asnd the midfielder held his head in his hands as it flashed inches wide.
Spartans were campeed out in their own half as Hearts continued to press and Lambikisa sent the ball in from the right and Carswell denied Vargas low down at the far post.
Vargas was then denied twice with shots from the edge of the box as the desperate Spartans rearguard worked overtime to keep the score level.
Lembriska had a chance out wide on the right but Carswell was at the near post to block but he could do nothing to stop defender Frankie Kent nodding home two minutes into injury time from a Cochrane corner to take Hearts through.
Spartans 1, Hearts 2.
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