Though Jamie MacDonald had to make an early save down to his left from a Charlie Adam drive, it was Pepe Reina who was busier in the opening stages.
The Spaniard was forced off his line to clear from the onrushing John Sutton before a full-stretch save to keep an Arvydas Novikovas effort from finding the top corner from 25 yards.
Sutton and David Templeton both had efforts just off target as Hearts' tactics of high-tempo, pressing football ruffled the English side's feathers.
The pace of Raheem Sterling down the left for Liverpool looked a potent weapon and Darren Barr almost knocked the ball past his own keeper as he slid in to prevent the winger getting a shot on goal.
And, as the visitors began to make their presence felt, Sterling stung the palms of MacDonald with a fierce effort as he cut inside onto his right foot.
Liverpool should really have taken the lead on 33 minutes. A lightning quick counter-attack was led by Jordan Henderson down the right.
His cross for Borini was inch-perfect, but with the goal gaping, the Italian slid the ball wide.
Hearts' threat had not evaporated, though, and Templeton fired a yard over the bar before the home fans and players made vociferous handball claims against Jamie Carragher, which were ignored by referee Florian Meyer.
Borini missed another glorious opportunity to give his side the lead after brilliant set-up play by Sterling, but with only the exposed keeper to beat, he struck the post.
The second half began at a slower pace than had been evident for most of the first period, but a vicious effort from a tight angle by Callum Paterson almost had the home crowd in delirium.
Adam and Novikovas swapped long-range efforts, neither of which troubled the respective keepers.
But with just 12 minutes remaining, the visitors made the breakthrough.
Kelly powered down the right, and though his cross was missed by Borini, the luckless Webster could not avoid the ball rebounding off him and into the back of the net.
Sterling was still looking lively in the closing stages, but on the two occasions he found room to shoot, his efforts failed to test MacDonald.
Paterson tried in vain to grab an equaliser, but Reina was equal to a powerful shot on the turn and the English side side emerged with a first-leg lead to take back to Anfield.
Liverpool manager Brendan Rodgers revealed afterwards that Andy Carroll missed the tie because of a hamstring problem, but hopes the striker will be fit for Sunday's visit of Manchester City.
He said: "Andy was due to start. He had a slight twinge on his hamstring. We had a scan on it and it didn't show a tear, just a slight disruption.
"We tried to give him 48 hours, let him travel and he trained yesterday to see how it was.
"It was still a bit tight and the medics' advice was it was too much of a risk for him to play."
Free kick awarded for a foul by Jordan Henderson on Daniel Grainger. Direct free kick taken by Daniel Grainger.
Callum Paterson has an effort at goal from long range that misses to the right of the target.
90:00 +1:04 Substitution
Substitution Fabio Borini goes off and Adam Morgan comes on.
The ball is delivered by Andy Webster.
Callum Paterson takes a shot from 12 yards. Save by Jose Reina.
Centre by Mehdi Taouil, Daniel Agger makes a clearance.
Shot by Andrew Driver. Save made by Jose Reina.
Substitution (Hearts) makes a substitution, with Scott Robinson coming on for Darren Barr.
Fabio Borini is ruled offside. Free kick taken by Marius Zaliukas.
Raheem Sterling takes a shot. Save by Jamie MacDonald.
Substitution Dale Carrick is brought on as a substitute for Arvydas Novikovas.
Shot from just outside the box by Raheem Sterling goes over the bar.
Substitution David Templeton leaves the field to be replaced by Andrew Driver.
Martin Kelly provided the assist for the goal.
77:08 Goal scored
Goal - Andy Webster - Hearts 0 - 1 Liverpool Andy Webster puts the ball into his own goal from 6 yards. Hearts 0-1 Liverpool.
Centre by Martin Kelly, Andy Webster gets a block in.
Stewart Downing produces a cross, clearance made by Marius Zaliukas. The ball is crossed by Stewart Downing, save by Jamie MacDonald.
Jonjo Shelvey challenges Mehdi Taouil unfairly and gives away a free kick. The free kick is delivered left-footed by Daniel Grainger from left wing, save by Jose Reina.
David Templeton produces a cross, clearance by Jamie Carragher.
Outswinging corner taken from the left by-line by Arvydas Novikovas, Jonjo Shelvey makes a clearance.
Ryan McGowan sends in a cross.
Joe Allen delivers the ball, Jamie MacDonald makes a save.
Martin Kelly sends in a cross, clearance made by Andy Webster.
Raheem Sterling delivers the ball, Marius Zaliukas manages to make a clearance.
Charlie Adam takes a shot. Jamie MacDonald makes a comfortable save.
Free kick taken by Arvydas Novikovas.
Substitution Jay Spearing goes off and Joe Allen comes on.
Free kick awarded for a foul by Martin Kelly on Arvydas Novikovas.
Daniel Grainger gives away a free kick for an unfair challenge on Charlie Adam. Charlie Adam restarts play with the free kick. The ball is delivered by Jonjo Shelvey, save made by Jamie MacDonald.
Fabio Borini fouled by Andy Webster, the ref awards a free kick. Daniel Agger takes the free kick.
The ball is delivered by Daniel Grainger, clearance by Jay Spearing.
Darren Barr handles the ball and concedes a free kick. Jonjo Shelvey takes the free kick.
Substitution Stewart Downing is brought on as a substitute for Jack Robinson.
Mehdi Taouil takes a shot. Blocked by Jay Spearing.
Raheem Sterling gives away a free kick for an unfair challenge on Ryan McGowan. Free kick crossed by David Templeton, clearance by Daniel Agger. Corner taken by Arvydas Novikovas from the left by-line, Daniel Agger manages to make a clearance.
Effort on goal by Arvydas Novikovas from outside the area goes harmlessly over the bar.
Charlie Adam produces a right-footed shot from outside the box and misses left.
The referee blows for offside against Raheem Sterling. Andy Webster takes the indirect free kick.
Shot from inside the six-yard box by Callum Paterson goes wide right of the target.
The ball is delivered by Ryan McGowan.
Corner taken right-footed by Jonjo Shelvey from the left by-line, save by Jamie MacDonald.
Fabio Borini is flagged offside by the assistant referee. Free kick taken by Jamie MacDonald.
Unfair challenge on Martin Kelly by Daniel Grainger results in a free kick. Direct free kick taken by Jordan Henderson.
Daniel Grainger crosses the ball, clearance by Daniel Agger. Ryan McGowan takes a shot from 25 yards. Jay Spearing gets a block in.
Free kick awarded for a foul by Charlie Adam on Mehdi Taouil. Mehdi Taouil restarts play with the free kick.
The referee blows his whistle to start the second half.
45:00 +0:10 Half time
Half Time The ref blows to signal half-time.
Fabio Borini has an effort at goal from deep inside the box hitting the post.
Charlie Adam takes a shot. Blocked by John Sutton. Charlie Adam takes a inswinging corner to the near post, Foul by Daniel Agger on Marius Zaliukas, free kick awarded. Direct free kick taken by Jamie MacDonald.
Inswinging corner taken left-footed by Charlie Adam from the right by-line, Jamie MacDonald makes a save.
Floated corner taken from the right by-line by David Templeton, clearance by Jamie Carragher.
The ball is swung over by Ryan McGowan, Shot on goal by Callum Paterson from just inside the area goes over the target.
Fabio Borini is caught offside. Jamie MacDonald restarts play with the free kick.
Foul by Fabio Borini on Mehdi Taouil, free kick awarded. Mehdi Taouil takes the direct free kick.
Foul by Jack Robinson on Arvydas Novikovas, free kick awarded. Free kick crossed by Arvydas Novikovas, clearance made by Jordan Henderson.
Jordan Henderson produces a cross, Fabio Borini produces a right-footed shot from deep inside the area which goes wide of the left-hand post.
Fabio Borini takes a shot. Blocked by Andy Webster. Charlie Adam takes the inswinging corner, Fabio Borini challenges Andy Webster unfairly and gives away a free kick. Jamie MacDonald restarts play with the free kick.
Callum Paterson delivers the ball, blocked by Jay Spearing. Outswinging corner taken right-footed by David Templeton played to the near post, Jonjo Shelvey manages to make a clearance.
Free kick awarded for a foul by Raheem Sterling on Ryan McGowan. Andy Webster restarts play with the free kick.
Raheem Sterling takes a shot. Save by Jamie MacDonald.
Raheem Sterling sends in a cross, Andy Webster makes a clearance. Corner taken by Jonjo Shelvey, Darren Barr manages to make a clearance.
Outswinging corner taken left-footed by Charlie Adam, Marius Zaliukas makes a clearance.
Charlie Adam takes the direct free kick.
Booking Darren Barr is booked.
Foul by Darren Barr on Charlie Adam, free kick awarded.
Raheem Sterling takes a shot. Andy Webster gets a block in. The assistant referee signals for offside against Raheem Sterling. Andy Webster restarts play with the free kick.
Free kick awarded for an unfair challenge on Jordan Henderson by Mehdi Taouil. Jay Spearing takes the direct free kick.
Fabio Borini gives away a free kick for an unfair challenge on Andy Webster. Andy Webster takes the free kick.
Shot on goal by David Templeton from just inside the area goes over the target.
Arvydas Novikovas takes the outswinging corner, Shot by Darren Barr. Jonjo Shelvey gets a block in.
Jonjo Shelvey is ruled offside. Jamie MacDonald takes the indirect free kick.
Effort from the edge of the area by John Sutton goes wide of the left-hand post.
Arvydas Novikovas has an effort at goal. Save by Jose Reina. Outswinging corner taken by Arvydas Novikovas from the left by-line, Charlie Adam makes a clearance.
David Templeton decides to take a short corner, The ball is sent over by Arvydas Novikovas, Charlie Adam manages to make a clearance.
Charlie Adam takes a shot. Save by Jamie MacDonald. Jonjo Shelvey takes the outswinging corner, Darren Barr manages to make a clearance. Corner from the right by-line taken by Jonjo Shelvey, Darren Barr manages to make a clearance. Jamie Carragher concedes a free kick for a foul on David Templeton. Daniel Grainger restarts play with the free kick.
Ryan McGowan concedes a free kick for a foul on Raheem Sterling. Charlie Adam takes the direct free kick.
John Sutton is flagged offside by the assistant referee. Jose Reina restarts play with the free kick.
Foul by Jonjo Shelvey on Mehdi Taouil, free kick awarded. Direct free kick taken by Mehdi Taouil.
The ball is swung over by Charlie Adam.
The match has kicked off.
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