Marouane Chamakh's 41st-minute header was all that separated the sides at Selhurst Park, lifting Palace off the bottom of the English Premier League table and within three points of safety.
Stewart Downing had a second-half effort ruled out for a foul in the lead up, a decision West Ham were far from happy with.
"Sam is still a bit livid after the performance, not just of our players, but the officials," West Ham assistant Neil McDonald said.
"We feel a bit hard done by with the goal that we scored that was given as a foul. That stopped us getting on the front foot and getting back into the game.
"We saw the replays and couldn't work out where the foul was. Decisions like that cost people points, we were on the front foot, we were attacking, we created a good chance and it's been chalked off.
"It's hard to take really. We felt it must have been for offside, but apparently not."
West Ham midfielder Ravel Morrison was yellow-carded in the dying stages and will miss Saturday's trip to Liverpool.
McDonald was annoyed by the suspension for Morrison's fifth booking of the campaign after the 20-year-old shoved Palace defender Joel Ward in the face.
"I've been told there was an incident but I wouldn't like to comment. I'm more annoyed that that's his fifth booking, so that's a suspension," he said.
"I think it's out of frustration that we didn't get anything out of the game."
