Gareth McAuley scored with their first effort on target, heading in Chris Brunt's 81st minute corner, before substitute Romelu Lukaku fired home an added-time second. Goalkeeper Ben Foster had kept out the hosts with a string of saves, including one from Steven Gerrard's penalty after Jonas Olsson was adjudged to have fouled Luis Suarez.
Rodgers did not attend his post-match press conference, restricting his opinions to his contracted obligations with broadcasters, but he refused to be downbeat and praised his side's showing.
"The players gave everything - it was one of them games we couldn't get the break through," the Northern Irishman told Sky Sports. "The longer it goes on it gives the opposition more encouragement.
"It was one of them games when we needed that first goal. We were on the front foot but it's a good away performance from West Brom. You come here, defend and as long as you can keep the score 0-0 you have always got that chance.
"I'm disappointed for the players. We have been on a terrific run especially at home, we have defended well throughout the season here."
Meanwhile, Baggies boss Steve Clarke, returning to Anfield for the first time since leaving in the summer as assistant manager after boss Kenny Dalglish was sacked, felt that was just reward.
"I was pleased with the way we set out from the first minute," he said. "We were aggressive and we tried to do the right things when we had the ball."
He went on: "For me, at this moment in time, Liverpool are playing as well as anyone in the league so to come here and get that result with the performance as well I don't think you can say we were lucky because we weren't.
"When you come to a place like Anfield they have got players who create chances so you know that at some stage your goalkeeper is going to have to make some saves. Unfortunately because of the referee he had to make an extra save - I am assured the penalty was harsh but I haven't seen it - but that was important for us as well."
The victory was the Baggies' first in the league since Boxing Day and Clarke was pleased for his players, who have had to deal with the backdrop of the Peter Odemwingie issue after the striker drove himself to QPR on deadline day in a failed bid to force through a transfer.
Odemwingie was not in the squad again and Clarke said: "It wasn't a difficult decision. I am delighted for the players. When things are going well you don't need to praise them because they are getting the results and the plaudits."
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- Steve Clarke