Reds full-back Trent Alexander-Arnold almost scored from a free-kick inside his own half early on before Sander Berge's 13th-minute penalty gave the visitors a lead for the first time this season.
The visitors threatened to extend their lead through Oli McBurnie, who fired wide, and Ben Osborn, who was thwarted by the returning Alisson.
But on 41 minutes Liverpool rallied, Firmino reacting quickest to score after Aaron Ramsdale had saved Sadio Mane's header.
Substitutes David McGoldrick and Oliver Burke both fired wide in the final 15 minutes as the Blades searched in vain for an equaliser.
A first victory in three matches lifts Jurgen Klopp's team to second place, behind leaders Everton on goal difference.
Sheffield United have one point from their six matches.
Budweiser King of the match: Sadio Mane (LIV)
Liverpool: 31 Oct v West Ham (H)
Sheff Utd: 31 Oct v Man City (H)