The Reds were superior for most of the game and they took an early lead in the ninth minute from the penalty spot after Luis Suarez was judged to have been tripped, Adam converted the penalty and Carroll added a second goal for the visitors moments before half time.
The result means the Reds have now won 13 of their last 14 matches against West Bromwich, who had come into this game unbeaten in their last four games.
Liverpool now climb into fifth place in the Premier League, as they keep into touch with the chasing pack, but they still trail league leaders Manchester City by ten points.
Liverpool who started the game without the injured Steven Gerrard and Jamie Carragher, began the game with speed and enthusiasm and they looked a much classier side than their hosts, while West Brom, who were without Shane Long, because of a knee injury, defended too deeply, were ineffective in attack and appeared disorganised in defence, this invited the Reds to attack.
West Brom's Jerome Thomas was alleged to have fouled Suarez for the penalty, which infuriated the home fans, but they could not blame the referee for their sides poor performance, in fact the home side failed to create a single opportunity in front of goal until the fortieth minute, when Peter Odemwingie could not connect with a cross from Tchoyi.
Liverpool continued to be the better team during the first half as Suarez and Carroll missed opportunities to increase the Reds lead, before Carroll scored the second goal of the game sliding the ball past West Brom goalkeeper Ben Foster.
The crowd became increasingly frustrated and aimed the anger at the referee as they still felt aggrieved following the earlier penalty decision.
The hosts started the second half more brightly as Somen Tchoyi saw his shot swerve narrowly wide of the Liverpool post, however the revival was short lived as Suarez and Adam proved to be a constant threat for the Reds, Foster also made good saves from Carroll and Jose Enrique to deny the visitors a third goal.
Stewart Downing also hit the post with an attempt at goal in the final minutes, as West Bromwich offered little resistance and made the victory pretty easy for the visitors from Merseyside.