Berahino's 44th-minute strike in the contest at The Hawthorns - his second top-flight goal, following up his winner at Manchester United in September - took his overall tally for the campaign to 10, including four goals for England Under-21s.
Talks between the 20-year-old, who is reported to currently be on £850 a week, and the Baggies over a new deal are ongoing.
And asked post-match for an update on that situation, Clarke said: "We are in negotiations with Saido's representatives and if I was the club, I would get it done as soon as possible.
"He has had a good season, and long may it continue."
Off the back of Berahino's rapid emergence, there has been talk of him potentially breaking into the full England squad in time for next summer's World Cup.
It is not talk welcomed by Clarke, who has been wary of the weight of expectation being placed upon the Burundi-born player.
"It is hard for me if you (the media) want to write about it," Clarke said.
"All I would say is that he is a young boy that, a few weeks ago, nobody was talking about.
"You have to keep the level of expectation down. In this country we are too quick to push people forward as the next big thing.
"For Saido, the most important thing is that he continues to develop as a player.
"That is what I am trying to do with him here and I think he is happy to work with me along those lines.
"My own personal opinion is that he is not quite ready for full international football.
"It is not for me to tell Roy (Hodgson, the England manager) how to do his job, but that is my personal opinion.
"People are trying to push him too far too quickly. Give the boy time to grow and develop."
Berahino, who had come on as a substitute just after the half-hour mark for hamstring-injury victim Billy Jones, swept the ball in from just inside the box having received it from Stephane Sessegnon's cut-back.
Gareth McAuley headed in Chris Brunt's corner in the 83rd minute to wrap up West Brom's win as managerless, rock-bottom Palace sunk to a ninth defeat in 10 league games this term.
The visitors created some notable opportunities, such as when Adlene Guedioura - who subsequently departed the field on a stretcher with damaged ribs following a collision with Boaz Myhill - saw an early shot saved by the Baggies goalkeeper, and when Mile Jedinak was only able to head wide from a good position in the second half.
Ultimately, though, the Eagles did not have enough, suffering a seventh straight loss in what was caretaker boss Keith Millen's second match in charge since Ian Holloway's departure by mutual consent.
The wait for Holloway's permanent successor to be confirmed goes on, and Millen said: "It is not an easy situation, with all the speculation about when the manager is going to come in and who it will be.
"The players have been focused - I don't think you can accuse them of not concentrating on the job in hand. But it is a tough time for everyone at the club really.
"The chairman is not going to rush into it. He needs to appoint the right person.
"But I think the sooner someone does come in, the better - then there will be new ideas, a fresh voice and hopefully a little bit of luck maybe that I don't think we are getting at the moment. Then things can change."
Asked if he knew when it might happen, Millen said: "No, I don't. I spoke to the chairman on Friday and he's busy doing what he needs to do.
"He told me to continue this week, so we are planning this week's training ready for the Everton game (next Saturday)."
Meanwhile, Clarke has confirmed that Victor Anichebe and Nicolas Anelka were left out of the Albion squad for the Palace match due to a hamstring problem and a virus respectively.
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