Dirk Kuyt was next to come close for Liverpool moments later, but his effort from inside the box drifted wide when he might have done better.
Brunt again came close for the away side after 14 minutes, but his strike from long range missed the target.
Brunt had the game's best chance just before the 30 minute mark, however, but was denied by a wonderful save from Reina in the Liverpool goal.
Suarez came close with a header for the home side late in the half, while Peter Odemwingie struck wide for the Baggies five minutes before the interval.
The home side started the second period the better, and Jordan Henderson struck wide from long range after 50 minutes, before Carroll tested Foster with a strike moments later.
Maxi might have opened the scoring with an effort after 57 minutes, but his strike cleared the bar by some distance.
Kuyt so nearly found the opener just before the hour mark in the match, but his effort from inside the box smashed off the post.
Liverpool continued to dominate the second period, and came close with a Carroll header, before Jay Spearing also missed the target after 65 minutes.
It was West Brom who took the lead, however, after 75 minutes when Odemwingie smashed the ball home from close range.
The result moves West Bromwich Albion into 10th in the Premier League table, while Liverpool remain in eighth.