Djokovic, inactive on the ATP Tour since losing to Jannik Sinner at the grass-court Grand Slam on July 10, won the first set 6-2 but then lost the next two 6-4 as he battled the heat and humidity in Ohio, even appearing to vomit at one stage.
The 24-time major champion won an 18-minute third game of the second set, fending off four break points over nine deuces, before taking a medical timeout, while he dropped to his hands and knees at certain points in a match that lasted two hours and 45 minutes.
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Tirante revels in 'best win of career'
Tirante, who can look forward to a third-round meeting with Spain's Martin Landaluce, made the decisive break at 4-4 in the decider after converting his fourth break point of the game and then closed out the win on serve.
He said afterwards: "This is the best win of my career. I really did a good job inside the court. I managed very well the nerves of playing against a legend like Novak. Now I feel relieved.
"It is all so much pressure when you cannot break [serve]. I had so many points to break and I couldn't, so that was the key. Keep believing in myself with my team and think about it after the match."
Djokovic was the third seed in Cincinnati despite sitting fifth in the world rankings, with world No 1 Sinner withdrawing due to a knee injury and world No 3 Carlos Alcaraz also absent as he continues to deal with a lingering wrist issue.
Top seed Alexander Zverev plays Britain's Cameron Norrie in the second round overnight.
Britain's Boulter out of women's tournament
British women's No 2 Katie Boulter saw her campaign end in the second round with a 6-3 6-3 defeat to Wimbledon champion Linda Noskova.
Noskova, who served 10 aces compared to Boulter's none, converted four of the nine break points she fashioned and was only broken once in return.
Boulter had snapped a five-match losing streak on the WTA Tour by beating America's Katie Volynets in three sets in the first round, winning the decider after losing the second set 6-0.
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