Murray's header gave Palace a fortuitous lead in the 41st minute before Scott Dann doubled their advantage in the 51st minute with a powerful header.
Murray, in the thick of the action all afternoon, added his second in the 63rd minute but was dismissed six minutes later for a second yellow card before Enner Valencia scored a 76th minute consolation goal for the hosts.
Eighth-placed West Ham have won just two of their last 12 league games since climbing to fourth in mid-December while Crystal Palace, up to 12th, have won four of their seven games since Alan Pardew joined the club last month.