Goals in either half from Roberto Firmino and Diogo Jota gave Liverpool a 2-1 comeback victory over a spirited Sheffield United at Anfield.
The Blades, with only a point to show from their opening five games of the season, took a surprise lead after 13 minutes when VAR ruled Fabinho's foul on Oli McBurnie had been just inside the home box - giving Sander Berge the perfect chance to beat the restored Alisson from 12 yards.
A stunning cross from Jordan Henderson teed up Liverpool's equaliser four minutes before half-time, with Sadio Mane's header palmed out by Aaron Ramsdale to Firmino, who scored just his third goal in his last 27 games for Liverpool.
The Blades continued to impress after the break without seriously threatening to further concern Alisson on his return from injury, and found themselves behind when Jota netted his first Premier League headed goal from Mane's excellent delivery.
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Victory sees Liverpool leapfrog Aston Villa - who have played a game less - into second place, level with city rivals Everton, who also have a game in hand on their neighbours.
Liverpool: Alisson (6), Alexander-Arnold (7), Gomez (8), Fabinho (5), Robertson (7), Henderson (8), Wijnaldum (7), Jota (7), Salah (7), Firmino (7), Mane (7).
Subs: Minamino, Milner (n/a).
Sheffield United: Ramsdale (6), Baldock (7), Basham (6), Egan (8), Stevens (6), Osborn (7), Berge (7), Lundstram (6), Ampadu (6), McBurnie (7), Brewster (5).
Subs: Burke (7), McGoldrick (6).
Man of the match: Jordan Henderson.
How Liverpool came from behind at Anfield
How would Liverpool cope without Virgil van Dijk? His name had dominated discussions in the build-up to Saturday's game, in spite of the Reds' relatively comfortable win over Ajax in midweek, but they flew out of the blocks at Anfield and could have led inside two minutes.
Andrew Robertson's inviting ball down the outside found Mane's run, and with Ramsdale opting to race off his line, the winger's slide pass towards Firmino looked certain to be tapped home until a vital sliding block from John Egan took it behind.
Trent Alexander-Arnold caused the goalkeeper more problems with an audacious free-kick from the half-way line which saw him back-peddling to tip the ball over his crossbar and spare his blushes.
The half would swing on a momentary lapse from Fabinho, so impressive in Amsterdam on Wednesday night, but rash to lunge in on McBurnie on the edge of his own box. Mike Dean initially awarded the visitors a free-kick, but VAR referee Andre Marriner ruled the foul had occurred inside the box, leaving Berge to beat Alisson from 12 yards.
Their sudden burst of confidence which followed was clear to see, as McBurnie then skewed an effort wide when well-placed before Ben Osborn's well-struck volley was palmed away by Alisson. The Blades had serious cause to believe they could have had another penalty with half-time approaching when Berge went over Diogo Jota's outstretched leg just inside the area, but a well-placed Dean saw nothing untoward.
- Alisson returned from injury for Liverpool as one of four changes from their Merseyside derby draw last weekend.
- Rhian Brewster made his first Sheffield United start in place of David McGoldrick, with Ben Osborn and Ethan Ampadu also brought in.
Moments later the half would take another sudden swing, as Liverpool made something out of nothing to level. Jordan Henderson's wicked cross from the right found Mane at the back post, but after keeping out his header Ramsdale was powerless to stop Firmino following up from close-range.
That didn't take the wind out of the Blades' sails before half-time, and neither did their heads drop after with George Baldock seeing a goalbound effort deflected behind within two minutes of the restart, before Salah fired just over at the other end from a half-cleared corner.
Soon Salah did find the back of the net with a textbook take and finish from Alexander-Arnold's whipped cross from deep, but his celebrations were cut short as a rapid VAR check ruled his run had taken him a foot or so beyond the last man.
They would not be cut short for long. Having defended so well against him all evening, Baldock allowed Mane space to cross from the left and Jota, having peeled off Enda Stevens at the back post, nodded in an excellent header.
Salah nearly wrapped up the points with another smart move nine minutes from time, turning Egan and heading for goal, but this time the woodwork, rather than VAR, denied the Egyptian.
Sheffield United never gave up the ghost in a performance which had plenty to savour but, most frustratingly for Chris Wilder, nothing to show for it as the Blades remained in the bottom three, while Liverpool moved joint top of the developing Premier League table.
What the managers said...
Liverpool manager Jurgen Klopp: "We started brilliantly, I liked the start a lot, we then conceded a goal which is not even a foul - that doesn't help. Then you are 1-0 down.
"For us it was a bit too hectic, when you have these four offensive players, but all four offer an option in behind, you have nobody in between. but then the goal we scored was perfect. We didn't do that often enough though, so we wanted to change that at half-time.
"We changed it at half-time, were completely superior, were everything we wanted to be, scored a second goal and another one which was I think offside, I don't know, but Sheffield Untied never gave up.
"They are a fighting unit, I respect it a lot, Chris is doing an incredible job, it's really tricky to play against them, but we won it and I'm more than happy."
Sheffield United boss Chris Wilder: "I thought the attitude was terrific. You're always going to be up against it because of the quality that you're playing against.
"They had more of the possession but I thought we were in the game right until the end. There's a lot of disappointed players in there and from my point of view, disappointed as a manager because I think we just did enough to get a result out of the game.
"It was more like us, it was more aggressive and more competitive. It was more of a Sheffield United performance that everybody has seen over the last two or three years and we needed to get back to that with the attitude in our approach.
"If you allow good players to get their heads up, they'll pick you off and I thought we made contact with them and then that gave us the opportunity to play as well."
Man of the match - Jordan Henderson
In addition to conjuring a rabbit out of a hat to create Liverpool's equalising goal when they looked likely to be frustrated until the interval by their visitors, Jordan Henderson led with the captain's armband, edging out Joe Gomez to the man of the match award.
He won possession back more than anyone else on his side (10) while making the most interceptions (3) of anyone on the pitch, and in addition to his superb cross for Mane, nearly provided the Senegal winger another golden opportunity with a wonderful pass into the area after half-time.
No-one came close to completing the number of opposition-half passes (66) as the skipper, who demonstrated his quality with and without the ball for the Premier League champions.
- Liverpool have won four consecutive Premier League games against Sheffield United by an aggregate score of 9-1.
- Liverpool extend their unbeaten run at Anfield to 62 games in the Premier League (W51 D11), winning 28 of their last 29.
- Liverpool are unbeaten in their last 13 Premier League home games when conceding the first goal (W10 D3), winning each of the last nine in a row.
- Sheffield United are winless in nine Premier League games (D1 L8), the longest current run of any side in the competition.
- Liverpool have conceded 14 goals in their first six Premier League games this season, it took them 15 league games to ship the same amount in 2019-20.
Liverpool are back in the Champions League this week, facing FC Midtjylland on Tuesday; Kick-off at 8pm. Next weekend they host West Ham at Anfield in the Premier League on Saturday Night Football live on Sky Sports Premier League from 5pm; Kick-off at 5.30pm.
Sheffield United face another tough test as they take on Manchester City at Bramall Lane next Saturday; Kick-off at 12.30pm.