The Reds needed an own goal from Andy Webster to defeat their Scottish opponents, who put up a good fight against a team that had them out-gunned on paper.
Hearts had several of chances of their own, with David Templeton and John Sutton a constant threat, but did they deserve more from the game?
Here, Sports Mole offers some analysis on the match.
On target: 6
On target: 7
Was the result fair?
A draw would have been a fairer reflection tonight. Although Liverpool saw more of the ball, Hearts played with more spirit and were the better team for lengthy spells in the second half. Brendan Rodgers's side looked far from clinical tonight, and Webster's own goal came against the run of play.
The Jambos did Scottish football proud, matching a team that are superior on paper for much of the game. It was a momentary lapse in focus that cost them the goal, and a similar error almost saw Liverpool break the deadlock at the end of the first half. David Templeton and John Sutton were an ever-present threat for Hearts, and home side had a pair of golden chances to draw level through Andrew Driver and Callum Paterson at the death, but it just wasn't their night.
Liverpool were rarely spectacular tonight, but they got the job done. Whenever Hearts made a mistake, the Reds were there to punish them. Fabio Borini should have put them in front seconds before the half-time whistle, and the Italian would have been on hand to capitalise had Webster not got the final touch when the goal when in. Raheem Sterling's performance was one of the biggest plus points for the Reds this evening. The youngster looked creative throughout and is certainly one to watch in the future.
Sports Mole's man of the match
Hearts' David Templeton showed a lot of spirit, causing Liverpool all kinds of problems. The Reds' defence did well to restrict him to long-range efforts, but his work rate was exceptional and he was unlucky not to get in among the goal.
Webster's own goal is the obvious one. Liverpool showed Hearts that those kind of lapses in concentration will be punished severely against Premier League opposition. Borini's sitter in the first half is a close second.
Florian Meyer didn't have a great deal to do tonight, though he got it right when it counted. Hearts had a couple of penalty appeals in the first half, which rightfully didn't stand, but it was a clean game otherwise.
Hearts have an away trip to Aberdeen to attend to before making the trip South of the border to Anfield for the second leg. They will have to come at Liverpool all guns blazing after conceding that away goal this evening.
Premier League champions Manchester City visit Anfield on Sunday when Liverpool will be looking to make up for a humbling opening day defeat at the hands of West Bromwich Albion. Then it's the return leg at against Hearts at Anfield.