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.