Cherki doubled City's lead in the 54th minute.
He picked up Foden's pass inside the box and weaved past several Palace defenders before finishing delicately beyond Dean Henderson with his boot's outside.
Yeremy Pino brought Palace back into the game with a stunning free-kick that hit the woodwork and deflected off Gianluigi Donnarumma for an own goal.
However, Cherki quickly restored City's two-goal advantage with Josko Gvardiol making the assist.
Haaland then sealed the victory late in the match, scoring his second of the night after being set up by Foden.
