The Brazilian nodded in a first-half winner to settle an unremarkable game and extend the Reds' unbeaten Premier League run to seven matches.
Jurgen Klopp's side climbed over Manchester City into third, having played a game more, to strengthen their claim on a Champions League qualification spot.
Firmino settled a poor match of few clear chances at the Hawthorns, although Nacer Chadli and James Milner wasted fine openings.
Albion have now taken just four points - against Manchester United and Arsenal - from their last seven games as their bid to break 50 points for the first time in the Premier League falters.
Despite their struggles, the Baggies began brightly and Hal Robson-Kanu directed Matt Phillips' effort wide but Liverpool gradually took control and nearly went ahead after 15 minutes.
Claudio Yacob squandered possession to Philippe Coutinho and he played in Firmino but the Brazilian dragged his angled shot wide.
The visitors bossed the early stages, although it was Albion who created the best opening when Chadli was left red-faced after 23 minutes.
Gareth McAuley found Phillips and he hooked the ball to the far post with Chadli waiting to tap in but the former Tottenham man missed his kick from four yards, having been put off by the slightest of touches from Firmino.
Immediately Coutinho volleyed wide at the other end and the lively Robson-Kanu fired straight at Simon Mignolet 10 minutes before the break but they were brief flash points in a tedious game.
Albion sat deep and denied Liverpool space, with their plan working until the Reds broke the monotony to grab the winner seconds before half-time.
Lucas flicked on James Milner's right-wing free-kick and Firmino stole ahead of Craig Dawson to head in.
It gave Liverpool impetus and they should have doubled their lead 11 minutes after the restart.
The visitors had maintained control and a precise build-up sent Firmino scampering into the area but Milner could only volley over his floated cross.
Divock Origi's header was disallowed for offside shortly afterwards and all Liverpool needed to win was to keep their concentration.
The Baggies offered little threat until 10 minutes from time when Mignolet needed to be alert to preserve the points.
Firmino, of all people, lost the ball to Salomon Rondon and he teed-up Phillips only for the winger to be thwarted by the onrushing Mignolet.
And substitute Alberto Moreno should have settled it in stoppage time but, after goalkeeper Ben Foster had come up for a corner and been dispossessed, shot wide of an open goal from 40 yards.