Much of the pre-match talk was about the hosts moving up to second with three points, although it was City that looked more like the promotion challengers in south London.
Nathan Redmond's excellent solo effort put the visitors ahead after 24 minutes with a Damien Delaney own goal doubling their advantage eight minutes later.
Palace looked resurgent at the start of the second half but Birmingham weathered the storm, with Ravel Morrison and Shane Ferguson then netting to secure a 4-0 win.
"When you come to a place like this you expect it to be tough and a close match," Clark said. "But not in my wildest dreams did I expect the margin of victory to be that big and, in all honesty, it was fully deserved as well. It wasn't a fluke. We had 11 fantastic performances. You can't come to a place like this and carry any passengers."
Birmingham's third win on the bounce moves them up to 10th in the standings and puts them within seven points of the play-offs.
Clark, though, is more concerned with things at the other end of the table as the season comes to a close.
"We are just looking at the clubs directly above us to try and peg them back," he said. "We are also not being arrogant or ignorant and thinking we are out of what is going on behind us as well.
"I am thinking those teams behind us will carry on their good form [on Saturday] and I am still in the thought process that it might be a record number of points needed to keep you in the division.
"We're well aware of everything around us in the division and I know it is a cliche but it is important we take it one game at a time."
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