Hilmer driver Quaife-Hobbs took full advantage from the start as Rossi stuttered away and third-placed starter James Calado (ART) failed to get off the line at all.
Julian Leal leapt into second for Racing Engineering ahead of Russian Time's Sam Bird, but both were hit by Stephane Richelmi at the first corner and lucky to escape damage.
As Richelmi pitted for a new front wing, and DAMS team-mate Marcus Ericsson also suffered a puncture, Quaife-Hobbs extended a 1s lead by lap four.
The track was then hit by a brief rain shower on lap nine, which caused both Leal and Bird to overshoot the Rettifilo chicane.
Rossi, who had recovered in fourth after his poor start, got a great run through Curva Grande and passed both of them with a brave lunge at Roggia.
The gap between the leaders fluctuated as the rain continued to fall sporadically, but Quaife-Hobbs held on, despite flirting with the gravel at the final corner, to win by 1.8s from Rossi.
Leal kept Bird at bay for third, with Bird's team-mate Tom Dillmann in fifth. Crucially, championship leader Fabio Leimer grabbed the two points for fastest lap on his way to sixth, which halved the points that Bird should have gained by finishing two places ahead of him, so the difference between them is now six points.
Rio Haryanto just missed out on fastest lap by 0.05s in seventh, having also saved his tyres until the end, with Johnny Cecotto Jr just grabbing the final point from Dani Clos.
Erstwhile series frontrunners Stefano Coletti and Felipe Nasr fought out a brilliant duel for 12th after their disasters yesterday, but again recorded zero points and have it all to do to overhaul Leimer and Bird.