Saints opened the scoring through Graham Carey's free-kick, before Craig Beattie cancelled that out when he headed his first goal for the Jambos.
Beattie wrongly had a goal ruled offside before setting up Rudi Skacel to head home just after the break, but Hearts were denied the victory when Nigel Hasselbaink's shot deflected inside the near post off Zaliukas.
Beattie was handed his first start for Hearts, while Stephen Elliott, Mehdi Taouil and Ryan McGowan also started at Tynecastle.
Scott Robinson missed out through illness, while Jamie Hamill was suspended.
The visitors were boosted by the return of Paul McGowan from suspension and Carey from injury.
Saints carved out the first real chance of the game early on when Gary Teale set up the shot for Steven Thompson and his tame effort was saved easily by goalkeeper Jamie MacDonald.
Teale then claimed his own shot was handled in the box by Danny Grainger but there was no penalty award from referee Stevie O'Reilly.
The Paisley side continued to push forward and McGowan rifled just past the post, before Hasselbaink's effort was tipped over by the goalkeeper.
Hearts had a decent chance of their own when Andy Driver found himself one on one with the goalkeeper but he was denied by an impressive save from Craig Samson.
Instead, it was Saints who took the lead after 27 minutes when Hasselbaink was fouled by Andy Webster on the edge of the box and Carey lashed through the wall and into the bottom corner from the resultant free-kick.
Hearts were back on level terms after 37 minutes when Grainger's corner picked out Beattie in the packed goalmouth and he sent a powerful header into the back of the net.
The Jambos made their first change a minute later when Adrian Mrowiec was replaced by Skacel.
Beattie had the ball in the back of the net again on the stroke of half-time but the effort was ruled offside, with television replays suggesting the assistant referee called it wrong and the goal should have been allowed to stand.
Hearts looked dangerous when they emerged for the second half and Elliott saw his shot saved by Samson. The ball broke for McGowan, whose effort was blocked, before Elliott eventually fired wide.
The home side raced into the lead three minutes after the restart when Beattie turned provider with a cross from the left for Skacel to nod home his 10th goal of the season.
Beattie should have had the net bulging again when he collected from McGowan and nutmegged Jim Goodwin only to blast over the crossbar.
Saints desperately tried to haul themselves back into the tie after such a promising start and they were back on level terms after 83 minutes.
Teale's thunderous shot was blocked by MacDonald and the ball broke for Hasselbaink.
The Dutch forward faced a tight angle and fired the ball towards the goalmouth and it deflected off the heel of Zaliukas and crept inside the near post.
With neither side able to find the decisive goal in the final minutes, both teams must do it all again on March 21 to determine who progresses to the last four.