Jordan Henderson put Liverpool ahead after eight minutes, scoring from Trent Alexander-Arnold's corner.
Rui Patricio denied Mohamed Salah twice early in the second half before Raul Jimenez equalised on 51 minutes, deftly heading in Adama Traore's right-wing cross.
Traore forced a save from Alisson before the Liverpool goalkeeper also denied Jimenez.
Firmino provided the decisive moment with six minutes remaining, firing into the top-left corner from 12 yards out for his eighth goal of the season, which have all come away from home.
Liverpool's 14th consecutive win puts them 16 points clear of Manchester City, having played a match fewer.
Wolves are seventh with 34 points.
Wolves: 1 Feb v Man Utd (A)
Liverpool: 29 Jan v West Ham (A)