As for West Brom, they have now suffered three defeats in a row, while their run without a goal has been extended to 375 minutes.
Liverpool went into the fixture in the West Midlands having under-performed during the first half of their visit to Stoke City last weekend, and it initially appeared that they were going to replicate their efforts from that match.
A number of passes went astray during the opening exchanges, and West Brom created the first chance when a miscued volley from Matt Phillips was almost diverted into the bottom corner by Hal Robson-Kanu.
Back up the other end, the Baggies wasted a golden chance when Phillips crossed for Nacer Chadli at the back post but he could not make a proper connection with his volley from six yards out and the opportunity was lost.
A minute later, Coutinho sent a difficult volley wide from the left-hand side of the penalty area, but another opening went begging for the home team as Robson-Kanu fired tamely at Simon Mignolet after collecting a pass from Phillips.
It appeared that the score would remain goalless until half time but with the last action of the first period, Firmino headed home from eight yards after James Milner's free kick had been flicked on by Lucas Leiva.
After the restart, Liverpool continued where they had left off and Divock Origi was quick to try his luck with a curling effort from inside the penalty area, but it went a few yards wide of Ben Foster's post.
Two minutes later, the visitors should have been two goals to the good through Milner, who was found in space at the back post by the impressive Firmino, but he directed his volley over the bar from close range.
Tony Pulis made each of his three allotted substitutions with more of a quarter of the game remaining, but he succeeded in disrupting Liverpool's rhythm at a time when they were threatening to put the result in no doubt.
As the contest moved towards the closing stages, Jake Livermore dragged a shot well wide of the post before Phillips squandered West Brom's best chance of the match but he was denied by Mignolet after good work from Salomon Rondon.
In added-on time, Liverpool should have made sure of the points after Foster was stranded upfield after coming up for a corner, but Alberto Moreno fired wide from 45 yards when he had teammates either side of him for the pass.
However, it did not prove costly to the visitors, who saw out the remaining minutes to claim a vital victory in the race for a Champions League place.