Late goals from Gareth McAuley and Romelu Lukaku - after a string of saves from goalkeeper Ben Foster, including a Steven Gerrard penalty, had kept them in it - saw the Baggies leapfrog their opponents into eighth.
But while Liverpool still hold out an increasingly fading hope of a top-four finish Clarke has only one objective on his mind - for now at least.
"We have never spoken about European competition. I said from the start '40 points, first target' and we are almost there," said the Scot.
"To be honest I thought we'd be beyond the 40-point mark by now and we would've had a few games left to attack the rest of the season.
"We need one more win to get 40 points and we have 12 games left.
"Hopefully we will get it quickly and then we'll see where we end up at the end of the season.
"We can only look from game to game. No-one was talking about European ambition before this game, everyone was looking at the teams below us and thinking where we were getting our next win from."
Victory ended a run of one point from six matches and was their first in the league since Boxing Day, ensuring they did a top-flight double over the Reds for the first time since the 1966-67 season.
"I thought we played well in the game and deserved to get the three points," added Clarke.
"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."
Got to love this guys humility. Liverpool should have promoted him instead of getting in Rodgers. It's hard not to respect someone like Clarke, it's exactly the type of personality a manager of LFC should have at the moment in my opinion. Rodgers is far from that!- dhirt11