Andrew Webster’s 78th-minute own goal was enough to give the visitors a slender lead ahead of the return match at Anfield next week.
And there was further good news for Liverpool as manager Brendan Rodgers said he expects to discover in the next 24 hours whether Real Madrid midfielder Nuri Sahin will choose Anfield over Arsenal’s Emirates Stadium as his preferred destination for a season-long loan move.
The manager was asked about a potential move in his press conference following last night’s victory.
He said: “Hopefully in the next 24 hours we’ll get some confirmation on that. It’s quite far [along the line]. He’s a very good footballer.”
Jamie Carragher praised the side’s character in the win last night, saying the victory shows Rodgers is already making a difference.
Carragher said: “The pre-season went really well and we obviously had a bit of a bump at West Brom so it’s nice to come here and get a result in a massive game.
“Obviously the new manager has come in and is finding his feet but it doesn’t need to be a slow process — we want it to be as quick as possible.
“It was a difficult game. We knew they’d come out flying and the atmosphere was fantastic. Great credit to Hearts and we’re pleased to come away with the result.”
Carragher was impressed with Rodgers’ youthful line-up, in particular striker Raheem Sterling, adding: “This club’s got a great tradition of bringing young players through and the way the manager wants to play suits him [Sterling] down to the ground.
“He’s come in and done really well and at the age of 17 it looks like he’s got a big future.”
In truth, the Edinburgh side can feel hard done by with the loss. The result was far more credible than Hearts’ 5-0 thrashing by Tottenham at the same venue last season and provides some hope ahead of the second leg at Anfield next week.
Andy Carroll failed to make the Reds squad — 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.
Rodgers opted instead for the goal threat to come from Fabio Borini and young Sterling.
Daniel Agger also started following his red card in the 3-0 defeat at The Hawthorns, but Steven Gerrard, Luis Suarez, Martin Skrtel and Glen Johnson were all left on Merseyside ahead of Sunday’s visit of champions Manchester City.
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 Jamie MacDonald.
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.
The visitors looked dangerous when Jordan Henderson delivered an inviting cross from the right into the box but Borini just failed to connect.
Hearts then claimed 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, who hit the post.
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 bar.
Midfielder Adam then had another go at goal but he was off target and succeeded only in smashing his shot off the advertising hoardings.
However, 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 found its way 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 Reina as Liverpool claimed a morale-boosting success.
HEARTS: MacDonald, Ryan McGowan, Webster, Zaliukas, Grainger, Paterson, Barr (Robinson 87), Taouil, Novikovas (Carrick 84), Templeton (Driver 79), Sutton.
LIVERPOOL: Reina, Kelly, Carragher, Agger, Robinson (Downing 62), Henderson, Adam, Spearing (Allen 67), Shelvey, Borini (Morgan 90), Sterling.
Ref: F Meyer (Germany).