Eden Hazard equalised for the away side after Shane Long had scored the opener for West Brom but Peter Odemwingie pounced five minutes after the restart to secure the points for Clarke's side.
"We were hanging on a little bit towards the end," he told Sky Sports News.
"Just after I made the third substitution, Youssouf Mulumbu flagged up he had a problem with a little bit of cramp in his hamstring, so we finished the game with 10 and a half players, if you like.
"It was just relief more than anything to earn the three points that I think the effort of the players deserved.
"(We have) character and we now look comfortable against the top teams. We've got enough about us now to defend well but to know when to play at the right times.
"There was a period in the first half when I thought we were knocking the ball around really well and in complete control of the game.
"As a coach, watching your players doing that against a top team like Chelsea, it is satisfying."
Clarke insisted nothing should be taken away from his side's victory, despite his opposite number omitting Juan Mata and Oscar from his starting line-up.
"It's not for me to comment. Chelsea have a squad of players they think is good enough to challenge for all the trophies and we've got a good squad of players here at Albion," added the Scotsman.
"To note any absences from the Chelsea team would be disrespectful to us. We went out there, performed well and got the win."
Clarke was also keen to praise Long, who turned in another strong overall performance in addition to scoring the opening goal.
"I have run out of words to describe Shane. He was up front on his own today and at times he was a little bit isolated," said Clarke.
"But he never stops trying for the team and his was one of a number of top performances for the team today."