Christian Benteke opened scoring for the visitors, before Mesut Ozil drew the Gunners level.
Wilfried Zaha and James McArthur extended Palace’s lead, before Pierre-Emerick Aubameyang pulled one back.
Arsenal now sit in fourth, a point behind third-placed Tottenham, but could find themselves fifth ahead of Wednesday’s trip to Wolves if Chelsea beat Burnley on Monday night.
Emery insisted, however, that the Gunners remain masters of their own top four destiny.
He told Sky Sports:
“Today we lost a very big opportunity but it doesn’t chance our idea.
“It is in our hands because we can continue with the possibility of winning and finishing in the top four.”