Both sides spurned first-half chances, with Fabio Borini the main culprit.
Hearts gave a good account of themselves but were made to pay for their own profligacy in front of goal when Webster put past his own keeper.
Martin Kelly fired in a dangerous cross and the Hearts defender looked on in horror as the ball rebounded off him and into the net.
On balance, it was harsh on Hearts, who had given as good as they got against their more-fancied opponents, who, despite not being at full strength, fielded a multi-million pound team.
Hearts were keen to avoid a repeat of the scenario at this stage of last season's competition, when they found themselves two goals down inside the opening quarter of an hour.
On that occasion the Scottish Premier League side were knocked out after a 5-0 first-leg defeat by Tottenham.
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 Brendan Rodgers' side emerged with a first-leg lead to take back to Anfield.