Hibs had chosen the occasion to parade the CIS Cup after the final whistle but the celebrations were hollow after the defeat.
It was a mistake by goalkeeper Andrew McNeil with just nine minutes left which proved decisive and give Hearts their first derby win at Easter Road for four-and-a-half years.
He failed to deal with a free-kick and slapped the ball into the path of Zaliukas, who accepted the gift.
In an untidy game, Hearts threatened first and McNeil saved superbly from Andrew Driver after just six minutes, but Hibs were soon into their stride.
Craig Gordon had to be at his best to thwart Scott Brown as the home side looked the more likely to open the scoring.
Zaliukas and Julien Brellier both picked up bookings as they sought to keep Hibs at bay.
Brown chose the wrong option when he was in the Hearts' penalty area and tried to cut the ball back for Abdessalam Benjelloun, when he should have gone himself.
For all Hibs' possession, Gordon was not over-worked in the Jambos goal in the first half.
Hearts carried some threat themselves and McNeil had to be on his guard to save from Hristos Karipidis two minutes from the break.
Guillaume Beuzelin headed over early in the second half from good position from a Merouane Zemmama free-kick.
Hibs sent Ivan Sproule on after an hour in a bid to break the deadlock but Hearts continued to frustrate them.
The Northern Ireland winger lashed one effort well wide when Steven Fletcher was well positioned inside.
But it was Hearts who snatched the win in the closing stages with Zaliukas celebrating his first Edinburgh derby goal.
Sproule had a late chance to salvage it, but Gordon saved and there was no Hibs forward there to tuck home the loose ball.