Medvedev is a former US Open champion and has reached a further five major finals.
While the 29-year-old is regarded as a hard-court specialist, his recent results at Wimbledon offered confidence he could improve on his first and second-round exits at the Australian Open and French Open respectively earlier this year.
But Bonzi dominated key tie-breaks in the first and third sets before racing clear to victory in an impressive four-set finish after three hours and seven minutes.
After congratulating his opponent, who celebrated with his team court-side, Medvedev slammed his racquets against his chair and his bag.
Bonzi, whose unexpected win ended his six-match Tour-level losing streak on grass, will play Australian Jordan Thompson next.
Elsewhere on Monday, Greek former world number three Stefanos Tsitsipas, 26, retired with injury after the 24th seed fell two sets down against French world number 113 Valentin Royer.