West Brom had taken just one point from six matches going into the trip to Anfield, and incredibly had not won any of their previous 18 top-flight away matches on Mondays, a run stretching back 44 years to March 1969.
Gareth McAuley scored with their first effort on target, heading in Chris Brunt's 81st minute corner, before substitute Romelu Lukaku fired home a second in stoppage time.
Before that 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.
Clarke, making his first return to Anfield since leaving during the summer following Kenny Dalglish's sacking, felt that was just reward.
"I was pleased with the way we set out from the first minute," he said after the game.
"We were aggressive and we tried to do the right things when we had the ball.
"I thought we played well in the game and deserved to get the three points.
"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.
"When he made that save from a penalty that should never have been given it gave us that positive belief and momentum to go and win the game."
Jonjo Shelvey, making his first league appearance since Boxing Day, played behind Suarez in the absence of injured Daniel Sturridge, and had the ball in the net early on - but was flagged for offside.
The closest Liverpool actually came to scoring in the first half was when Steven Reid's attempted left-footed clearance of Shelvey's low cross cannoned off his standing leg and looped over the crossbar from four yards.
The home side struggled for fluidity, even after bringing on Raheem Sterling and Fabio Borini, before Suarez went down under Olsson's attentions - but Foster blocked the penalty to keep the scores level.
Liverpool were made to pay when hit with a sucker punch nine minutes from time, after Brunt swung over a corner from the right wing. McAuley had a free run in the area to power home and put the visitors ahead.
Substitute Lukaku exploited Liverpool's pressing, hitting the home side on the break to surge past Daniel Agger and finish off the game - a win which moves Albion a point above Liverpool in the English Premier League table.
Brendan Rodgers elected not to attend his post-match press conference, restricting his opinions to his contracted obligations with broadcasters where he tried to extol the positives from the performance.
"The players gave everything," he told Sky Sports. "It was one of them games we couldn't get the break through.
"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."
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