It took City just 12 minutes to go into the lead with Aaron Mooy turning his marker before finding the back of the net, and midway through the opening period, they had doubled their advantage.
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Melbourne had opportunities to make it three, with Mooy curling just wide from a free kick and Williams firing a strike off target, and after the restart, the home side were made to pay for failing to put enough distance between themselves and their opponents.
The momentum was with the visitors, and 11 minutes from time, David Carney's free kick was nodded into the net by Trifunovic for his third goal of the season.
The result sees Newcastle go top of the A-League standings, while Melbourne remain in fifth position.