Melbourne took the lead just before the break when Aaron Mooy was played in down the right and the forward held his composure to fire the ball through the legs of Birighitti from a tight angle.
City extended their advantage in the second half as Ben Garuccio raided forward and teed up Fornaroli inside the box, which allowed the forward to poke his effort through Birighitti and into the net.
One minute later, the visitors notched their third goal with a break down the right as Ivan Franjic raided down the wing and sent a low cross into the box for Mooy to convert from close range.
Melbourne secured their victory when Fornaroli notched his side's fourth goal as he was played in behind and weaved his way past Lee Ki-Je before slotting his effort into the net.
The result lifts City into fourth place in the table, while Newcastle remain in seventh.