Roberto Firmino and Diogo Jota were on target for Liverpool, who had fallen behind to a Sander Berge penalty.
Liverpool returned to winning ways in the Premier League as Diogo Jota's second-half goal secured a 2-1 win over Sheffield United at Anfield on Saturday.
A controversial draw with Everton last time out followed a 7-2 humiliation at the hands of Aston Villa before the international break and Jurgen Klopp's side looked set for more disappointment when a Sander Berge penalty put the Blades ahead after 13 minutes.
Robert Firmino's first of the season drew them level ahead of the interval, though, before Jota sealed all three points in the 64th minute with his second goal for the club following his move from Wolves last month.
The result moved Liverpool up to second, level on points with leaders Everton, while Chris Wilder's side - who have failed to win so far this season - remained in 19th.
Liverpool almost went ahead inside the opening five minutes but Aaron Ramsdale backtracked well to tip over Trent Alexander-Arnold's audacious free-kick from 45 yards.
The hosts then conceded their first penalty at Anfield in the Premier League since October 2018 when Fabinho was deemed to have brought down Oliver McBurnie just inside the area, with Berge stepping up to coolly slot past Alisson.
The Liverpool goalkeeper kept out Ben Osborn's powerful volley soon after as Wilder's side threatened to double their advantage.
It was the hosts who scored next, though, Firmino slotting home four minutes before the break after Ramsdale parried Sadio Mane's header.
George Baldock and Mohamed Salah exchanged half-chances at the start of the second period before the latter saw a fine effort ruled out for offside shortly after the hour mark.
The Reds' disappointment did not last long, however, with Jota superbly heading home Mane's left-wing cross to put the hosts ahead.
Salah clipped an effort against the post with 10 minutes remaining but it mattered little as Klopp's men ultimately held on with the minimum of fuss to extend their unbeaten Premier League run at Anfield to a whopping 62 games.
Reds made to sweat by plucky Blades
Liverpool avoided making it three Premier League games without a win for the first time since May 2018 but they did not have it all their own way against a spirited United, who had 13 shots on goal.
Without the injured Virgil van Dijk for the first time in 95 top-flight games, the Reds could easily have been 2-0 down before Firmino pounced late in the first half.
While they ultimately came through unscathed, there was enough vulnerability on show to suggest there will be some difficult tests ahead for Klopp's men.
