Firmino - back in the Reds’ starting line-up along with Philippe Coutinho - converted James Milner’s free-kick, via a flick-on from Lucas Leiva, in stoppage time at the end of the first period to move Jürgen Klopp’s side back into third place in the standings.
The visitors had the better of the second half, too, but were unable to add to their advantage. Milner blazed over the crossbar from close range before Divock Origi saw a close-range header disallowed due to Firmino straying offside shortly after the hour.
Simon Mignolet denied Matt Phillips as West Brom broke away with 10 minutes to go, and Liverpool stood firm in the face of some late pressure to claim what could prove to be a vital three points in the race for Champions League qualification.