The result at Anfield means that Brendan Rodgers' side go through 2-1 on aggregate following their 1-0 victory in the first leg in Scotland last week.
The first chance of the game fell to Hearts as early as the fifth minute, when Callum Paterson tried a half-volley from outside the Liverpool penalty area, but he failed to find the net.
However, the hosts began to dominate proceedings thereafter and could have drawn first blood three minutes later through Adam Morgan.
Liverpool crafted another good chance on 14 minutes when Stewart Downing, Morgan and Luis Suarez all combined, but the Uruguayan striker's header was eventually cleared.
Martin Kelly and 18-year-old Morgan both asked questions of the Hearts defence before Liverpool captain Steven Gerrard went solo but without success in the 29th minute.
Four minutes later Morgan found the target but the assistant referee ruled out his goal as the ball had gone out when Suarez delivered the cross.
Much against the run of play, Hearts could have taken the lead moments later when David Templeton carved out a brilliant cross from the right, but John Sutton failed to keep his close-range header on target.
Templeton was active again in the 39th minute when he tried his luck from outside the Liverpool 18-yard box, but could not keep his shot on target.
Liverpool continued to be the dominating side in the second half, with Jonjo Shelvey shooting just over the Hearts crossbar five minutes into the restart.
Gerrard came close for the home side in the 68th minute and six minutes later, Martin Skrtel had a free header tipped over by Jamie MacDonald, although the centre-back was deemed offside.
Hearts levelled the aggregate score with five minutes left on the clock, as Pepe Reina spilled David Templeton's speculative shot from outside the box over his line.
There was yet more drama to come at Anfield, as Suarez shot home past MacDonald in the 88th minute from the left.
Liverpool held on for the remainder of the game to earn a 1-1 draw in the second leg and a 2-1 aggregate victory in the tie.