Crystal Palace took a first-half lead, despite being outplayed for long periods, through debutant Cenk Tosun. City then saw a penalty decision overturned by VAR.
Sergio Aguero's 250th goal for the club on 82 minutes levelled the scores.
And five minutes later the brilliant Argentine headed home his second of the game to put City ahead, but Palace had the final say as Fernandinho deflected a cross into his own net on 90 minutes.
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