Melbourne dominated much of the game at AAMI Park and were unlucky not to go in front in the first half when Alex Terra saw a 25-yard drive hit the post.
Central Coast keeper Matthew Ryan was also in inspired form, denying the hosts on a number of occasions before Adam Kwasnik snatched victory when he converted Michael McClinchey's delivery a minute from time.
Meanwhile, Adam Hughes' late strike denied Sydney FC all three points and extended Perth Glory's unbeaten run to five games with a 1-1 draw at Campbelltown Sports Stadium.
Sydney went in front just before the interval when Mark Bridge got on the end of Nick Carle's cross.
Carle then missed an open goal six minutes into the second period and Sydney were made to pay two minutes from the end as Hughes headed home Evan Berger's corner.