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.
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.
Player ratings
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.
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