The American Kocinski grabbed a dramatic victory in the opening dry race, getting the better of a seven-way battle for the lead in the 18-lap encounter to beat his Honda team-mate Aaron Slight by a wheel's length in a 190 mph sprint to the line.
The championship leader, Carl Fogarty, took a close third with the first three separated by only 0.008sec.
Thunderstorms before the second race helped the challenge of Chili, who grabbed the lead on the 10th lap and eased away, opening up a six-second lead before finally crossing the finish line 1.8sec ahead of Kocinski.
Jamie Whitham revelled in the treacherous conditions on his Suzuki to take his second third place in successive rounds, while Fogarty finished fourth.
Slight finished fifth, after colliding with Akira Yanagawa while the pair lapped a slower rider. Kocinski's win was his third of the year to equal Slight's tally, while Chili's win was his third in three years at Monza.
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