The result leaves Melville in eighth spot in State League Division Two after their first five games.
In a tight first half, both teams struggled to gain the upper hand.
Curtin’s Arijan Stovrag came closest to breaking the deadlock when he found space on the edge of the box, but his shot whistled wide.
In the second half, Melville failed to clear its defensive lines and Kahled Salhah pounced to give the visitors the lead inside a minute.
Get in front of tomorrow's news for FREE
Journalism for the curious Australian across politics, business, culture and opinion.
READ NOWCity’s response was swift, as Mark Lucas whipped in a free kick and Cormack Abbott’s deft header evaded Jack Kenny in goals and drew them level.
Melville pressed for a winner and almost took the lead when some great work by Louie Granich ended in Kenny denying Simon El Chami from point-blank range.
Minutes later, the away side re-took the lead when Stovrag caught Grgurinovic off his line and chipped the ball into the net.