John McGlynn’s men more than held their own, albeit against a weakened Liverpool side, and did more than enough to deserve to have taken the tie to Anfield next week all square.
The only goal of the game came 12 minutes from the time, the Hearts defender turning the ball past his own keeper from a fizzing Martin Kelly cross.
Hearts now have it all to do on Liverpool’s home ground in the second leg next Thursday night as they travel there knowing that they need at least two goals to have any hope of progressing.
Jamie MacDonald made it into the starting line-up after the birth of his daughter yesterday, leaving new signing Peter Enckelman on the bench.
Liverpool left a host of stars at home, Steven Gerard, Luis Suarez and Martin Skrtel among those who were rested, Raheem Sterling and Jack Robinson brought into the first 11 by Brendan Rogers.
The opening minutes were frenetic as both sides settled into their rhythm and Ryan McGowan, who was given the right-back slot, looked as though he would have a hard shift against the lively Sterling.
A roar of anticipation went round Tynecastle with seven minutes gone when a loose ball saw John Sutton chase in on goal but Pepe Reina kept a cool head to clear before the Hearts striker could get a touch.
Hearts almost bagged the opener in 11 minutes when some brilliant play from Mehdi Taouil set Arvydas Novikovas up and the Lithuanian let one fly from 20 yards.
It was dipping into the top left-hand corner of the net, only for Reina to somehow get up to his right to tip it over with one hand.
The home support sensed that their side might just be able to do something against their new-look Anfield opponents and the impetus was certainly with McGlynn’s men.
Darren Barr was booked with 23 minutes gone for a pull and late challenge on former Blackpool midfielder Charlie Adam.
Callum Paterson was having a good game, as was McGowan and when the two linked up in 27 minutes it looked like the Australian would get a shot in on goal only for him to take one touch too many and put the ball too far in front of himself.
Despite the excitement of the last few days, MacDonald looked focused and alert and did well to get both hands in front of himself to block a curling effort from Sterling on the half hour.
Charlie Adam bawled at his team-mates telling him that they had to do better – and he was right.
Taouil put his side in trouble ten minutes before the break when his poor header gifted possession to Jordan Henderson and the Liverpool player picked out Fabio Borini in the middle but, fortunately for Hearts, his shot trickled just wide.
The Hearts supporters shouted for a penalty just before the break when Webster’s header downward appeared to be handled by Carragher but the referee waved play on and the Liverpool defender claimed that the ball had come off his torso.
Liverpool should have gone in one up at the break though when another poor ball from Taouil was picked up by Sterling and played into the path of Borini but he could only smash his effort off the base of MacDonald’s left-hand post.
That was a real let-off for Hearts, who had allowed some slackness to creep into their play.
There were no changes for either side at the break but the Jambos started the second 45 in determined fashion.
Paterson pushed forward in 55 minutes and managed to get a shot in from the byline but it was just too tight an angle for the youngster, then it was Novikovas’ turn to have a go from distance and fire just over the bar.
Hearts were more than holding their own, McGowan beating Adam down the right, only for his cross to be too deep and then Novikovas playing the ball off Martin Kelly for a corner.
But the Tynecastle side shot themselves in the foot with 12 minutes left, a cross from the right from Kelly being turned into the net by Webster.
There wasn’t much that the big defender could have done to avoid the ball but it was still a real kick in the teeth for The Jambos, who had been more than deserving of a draw over the course of the 90 minutes.
Hearts: MacDonald, Grainger, McGowan, Barr (Robinson 88) Webster, Templeton (Driver 78), Sutton, Taouil, Novikovas (Carrick 84), Zaliukas, Paterson. Unused subs: Enckelman, Smith, D McGowan, McHattie.
Liverpool: Reina, Agger, Henderson, Spearing (Allen 68), Carragher, Adam, Borini (Morgan 90), Sterling, Shelvey, Kelly, Robinson (Downing 61). Subs: Jones, Coates, Leiva, Flanagan.
Referee: Florian Meyer