The Edinburgh side impressed for much of the tie before suffering a late blow against their Barclays Premier League visitors, who made the trip across the border without a number of star players.
The result was far more credible than Hearts' 5-0 thrashing against Tottenham at the same venue last season and at least provides some hope ahead of the second leg at Anfield next week.
Andy Carroll failed to make the Reds squad for Thursday night's clash - despite his presence at the stadium - as a question mark remained over his Liverpool future, with the £35million striker linked with a return to Newcastle.
Boss Brendan Rodgers opted instead for the goal threat to come from Fabio Borini and Raheem Sterling.
Daniel Agger also started following his red card in the 3-0 defeat at West Brom, but Steven Gerrard, Luis Suarez, Martin Skrtel and Glen Johnson were all left on Merseyside ahead of Sunday's visit of champions Manchester City.
Goalkeeper Jamie MacDonald was between the sticks for Hearts - agreeing to play after his partner gave birth to a baby girl on Wednesday - meaning Peter Enckelman had to settle for a place on the bench rather than his Jambos debut.
The home side was boosted further by the news that skipper Marius Zaliukas was able to start after recovering from a dead leg.
The first chance fell to Liverpool when Charlie Adam embarked on a surging run that took him to the edge of the box before unleashing a low drive that was blocked by MacDonald.
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At the other end, Hearts had a decent chance of their own to open the scoring when a long range effort from Arvydas Novikovas forced a great save from Pepe Reina.
John Sutton then dragged his shot inches wide of the post and David Templeton fired just over the crossbar as the home side continued to threaten in front of goal.
After the flurry of chances for Hearts, Liverpool gradually came back into the game without causing too many problems for the defence, until a swerving shot from Sterling was punched to safety by the goalkeeper.
The visitors looked dangerous again when Jordan Henderson delivered an inviting cross from the right into the box but Borini just failed to connect in front of goal.
Hearts then claimed Jamie Carragher handled a Templeton corner in the box but the pleas for a penalty were ignored by referee Florian Meyer.
Liverpool should have been ahead just before the break when Mehdi Taouil was dispossessed by Sterling and he set up Borini for a shot that appeared net-bound but crashed off the post instead.
Hearts, bolstered by their performance so far, were not too far away after the restart when Callum Paterson's powerful angled drive dipped just over the crossbar.
Adam then had another go at goal but he was off-target and succeeded only in smashing his shot off the advertising hoardings.
The breakthrough came with 12 minutes to go and Hearts fell behind in cruel circumstances when an excellent Martin Kelly ball from the right was missed by Borini but came off the shin of Webster and into the back of his own net.
Substitute Andy Driver could have snatched a leveller a couple of minutes from time but his effort was smothered by the goalkeeper.