The English League Cup winners won the battle over the impressive Scottish Cup champions in Edinburgh, pushing the Reds all the way, with Sterling looking good for the visitors
The hosts went toe-to-toe with Brendan Rodgers' side in a very bright first half in front of a buzzing Tynecastle crowd. Arvydas Novikovas went closest for the Scottish outfit as they put real pressure on the visitors, but some good work from Raheem Sterling saw Fabio Borini waste the best chance of the 45 when he hit the post despite being clear through on goal.
The second-half saw far less action than the first, with neither defence allowing the opposition to create clear chances. At times the Reds looked like they were on the verge of shifting up the gears. They failed to cause the hosts any major problems, but were granted a winner when Andy Webster put the ball in his own net.
John McGlynn started with John Sutton as a lone forward, with David Templeton supporting from midfield. Meanwhile, Arvydas Novikovas and Callum Paterson lined up on either flank.
Liverpool travelled to Tynecastle without the likes of Steven Gerrard, Luis Suarez, Martin Skrtel and Glen Johnson, with Rodgers handing starts to youngsters Jack Robinson, Jonjo Shelvey, Jay Spearing and 17-year-old Sterling.
There was no place for Andy Carroll in the first XI or the bench, but Charlie Adam, Jordan Henderson and Borini were given chances to impress, with Pepe Reina, Martin Kelly, Daniel Agger and Jamie Carragher also selected.
The Reds looked to pass the ball out from the back early on in the typical Rodgers style, but were sent a warning sign when a weak Carragher back-pass nearly gifted the hosts a dream start, with Reina forced to rush out of his area to clear.
Spurred on by the immense atmosphere created from the home supporters in Edinburgh, Novikovas fired a bullet towards the Reds' goal from range after some brilliant work by Mehdi Taouil, which the Spanish keeper was forced to push wide.
Sutton was the next Hearts man to have shot at the visitors' goal, as the ten minute mark passed, he turned swiftly to work a yard of space and get free of Carragher but his strike just swerved away from goal. Liverpool were firmly on the back foot, however.
Hearts' ambition and positivity was highlighted by a corner after 16 minutes, Novikovas delivered the set-piece to the edge of the area where Darren Barr was waiting to attempt a first-time volley - the shot didn't reach the goal but their confidence was evident. Then, David Templeton unleashed an effort with his left after a driving run down the flank, the ball travelling at pace across goal but off target.
Sterling was given his first real chance to show what he can do after 20 minutes, Spearing found the winger with a terrific ball out to the right, and he raced free on the flank before cutting in and riffling a shot at goal but saw his strike blocked. The youngster then caused more panic as he looked to run in behind the hosts' defence, with Webster almost finding his own net as he was forced into putting the ball behind.
Borini then had a good chance to give Liverpool the lead with ten minutes left in the half, after a spell where neither side could find the right final ball. The former Roma loanee shipped the ball out wide to Henderson before carrying on his run. The midfielder then played a lovely pass over the top towards the Italian who just failed to make contact with his attempted volley as the ball came over his shoulder.
Five minutes later Hearts once again put their foot on the gas. Templeton smashed a volley just over from the edge of the area, before the home side had a penalty appeal, correctly, turned down after they claimed handball against Carragher.
Down the other end, however, Liverpool should have taken the lead just before the break. Sterling's quick thinking led him to intercept at attempted cross-field ball by Taouil. Bringing it down with instant control, the 17-year-old galloped clear before laying the ball across to Borini, who slotted the ball past the outrushing keeper but against the post.
The tempo dropped after half-time with both sides struggling to find any rhythm. It was Sterling who was first to threaten. After playing a one-two with Adam the winger dashed into the Hearts' box before being brought down and appealing for a penalty, but nothing was given. Moments later, the Scottish side's own 17-year-old Paterson lashed a volley wide at the other end.
As the half went on both sides failed to create any major talking points. The Reds made their mark on possession and looked to slightly dominate with the ball, but could find no way through the Hearts' back line.
Liverpool did eventually get the breakthrough with just over ten minutes remaining. Kelly got forward well from his right-back position and fired in a dangerous low cross which, under pressure from Borini, Webster put into his own goal.
Hearts weren't done yet though. Firstly Driver's shot on the turn looked dangerous but ended up with a routine save for Reina, and then Paterson forced the Spaniard into a very good low save as his right-footed strike looked destined for the far corner, the keeper just tipping it wide to ensure his side claimed victory.