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Djokovic, a three-time Cincinnati champion and former world number one, made an impressive start, winning the first set comfortably.
However, the 39-year-old gradually showed signs of physical discomfort as the match progressed, with temperatures reaching 29°C and humidity at 72%.
The Serbian received a medical timeout during the second set, after which Tirante increasingly took control of the contest.
The 25-year-old Argentine relied on a powerful serve and aggressive groundstrokes, amassing 13 aces and 40 winners en route to victory.
The match was Djokovic's first since his defeat by Jannik Sinner in the Wimbledon semi-finals a month ago.
Currently ranked fifth in the world, Djokovic has won a record 24 Grand Slam titles.
The defeat marked his first second-round exit in Cincinnati since 2007 and occurred just two weeks before the US Open, the season's final Grand Slam tournament.