The day belonged to Etixx-QuickStep's two Italians as they outfoxed compatriot Moser at the end of the 240km stage that began in Muggio.
It was Trentin's maiden Giro stage victory.
Brambilla, who led the event after winning stage eight, was ahead of Moser over Pramartino, a category two climb and the stage's steepest summit, but ended up third as his team picked up their fourth stage win of this year's race.
German Marcel Kittel, who also rides for Etixx-QuickStep, notched back-to-back victories in the second and third stages.
Trentin, who has twice won stages on the Tour de France, was part of a small group of riders chasing the two leaders from an initial 24-man break that formed early in the day.
He began closing the gap at San Maurizio, five kilometres from the end, and joined the leaders on the home stretch before Trentin surged past to seal a memorable victory.
(Reporting by Simon Jennings in Bengaluru; Editing by Tony Jimenez)
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