After an upset in the opening fixture of Group B fixtures when number two seed Karen Khachanov was beaten 3-1 by fellow Russian Daniil Medvedev - the tournament is played in a best-of-five sets format but only four games is needed to win a set - it looked as if world number 37 Rublev could also slip up against Quinzi, who came through a local qualifying tournament which awarded a wild card to an Italian player.
Quinzi - a former Junior Wimbledon champion and world number one in 2013 but currently ranked 306 on the ATP seniors circuit - stormed through the opening set 4-1.
Rublev, who reached the quarter-finals of the US Open, recovered to win the second set 4-0 in 15 minutes and then moved ahead after securing the third set tie-break 4-3 (7/3).
However, left-hander Quinzi made sure the Fiera Milano crowd, who were allowed to move around during the match as part of the innovations being trialled at the event, got full value by forcing a decider after an early break in set four, which he went on to take to love.
Eventually Rublev, winner of his first ATP title at the Umag Open in Croatia earlier this year, came through 4-3 (7/3) in the final-set tie-break, Quinzi having twice saved match point.
Rublev is third-favourite in Sky Bet's outright odds at 5/1.
Earlier, Medvedev, seeded seventh, had pulled off an unexpected victory over Khachanov, the world number 45, when he came from behind to win 2-4 4-3 (8/6) 4-3 (7/3) 4-2.
Elsewhere in Group A, Hyeon Chung of South Korea recovered from losing the opening set to beat 18-year-old Canadian third seed Denis Shapovalov, the youngest player in the eight-strong field, 1-4 4-3 (7/5) 4-3 (7/4) 4-1.
In Tuesday's other Group B fixture, fourth seed Borna Coric of Croatia coasted past Jared Donaldson in straight sets, defeating the American 4-3 (7/2) 4-1 4-3 (7/5).
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