Exeter took the lead through Guillem Bauza but goals from Jermaine Easter, with a penalty, and Kagisho Dikgacoi meant the visitors took the spoils.
Exeter got off to the perfect start with a goal inside the opening two minutes.
Bauza volleyed Liam Sercombe's corner goalwards and John O'Flynn flicked a header over Palace goalkeeper Lewis Price and into the net after two minutes.
However, Palace equalised in controversial fashion after 24 minutes when Exeter's Tom Doherty was penalised for handball.
Easter made no mistake with the resulting penalty.
Palace went in front on the stroke of half-time when Wilfred Zaha beat Exeter goalkeeper Artur Krysiak to a loose ball and crossed to the back post where Dikgacoi headed into the empty net.
Exeter had their chances to level with O'Flynn and Alan Gow going close.
But the home side did not have enough to breach a well drilled Palace defence.
Palace could have added to their tally, but Glenn Murray missed the best of their chances after being set up by the dangerous Zaha.