The Baggies had failed to find the net in their opening three games of the campaign, but Gareth McAuley's late header secured a point.
Clarke made three second-half changes as he attempted to overturn the one-goal deficit and the former Chelsea coach says he thought that the performance merited at least a point.
"I think the way we played, we deserved to get something," he told Sky Sports.
"Obviously when you're going into injury time and you're still one down and you haven't scored your first goal of the season, you start to get a little bit anxious.
"But I said to someone back on the bench that I fancied us to score before the end and fortunately, a great delivery and Gareth does what he does best - attacks the corners.
"Goals-wise, Gareth's a threat in the box. We've got good set-play delivery and boys who can attack the ball there so we're always a threat from set-plays.
"What was disappointing for us was that we lost one from a set-play so that's unusual, it's not like us. But like I say, I thought the team performance today was better, we played well, we stayed in the game.
"Second-half we played some good football, created some decent chances and on another day, maybe with the confidence a little bit higher, we would have got the goal from open play, but that will come in the future, I'm sure.
Clarke also spoke of the difficult task he faced as he attempted to gel some of his new signings together following the international break.
"When you make a number of changes right on the deadline day, it's difficult enough, but when they go away for 14 days or whatever it was to be with their countries, you have to find a way quickly," he said.
"But when you have good players, it makes it a lot easier."
Deadline day signings Victor Anichebe and Morgan Amalfitano were handed their West Brom debuts and Clarke believes that they both proved their worth at Craven Cottage.
"Victor is a big strong striker, he showed all his qualities today. He can hold the ball up, can link the play well. There was some good understanding with Nicolas (Anelka) as the second striker," said Clarke.
"Morgan's come from the top division, a top team in France. He understands the game, he understands his position very well and I thought for his first game in the Premier League, he's got to be really pleased with his contribution."