After missing out on match point in the second-set tie-break, Alcaraz found himself 5-3 down in the third, but he saved two match points to pull back to 5-4 before breaking Djokovic's serve to level the set.
However, Djokovic went 5-4 up on Alcaraz's serve in the tie-break before winning the next two points to claim victory in three hours and 49 minutes - the longest best-of-three-sets final in ATP Tour history.
"This rivalry just gets better," Djokovic said. "It gets better and better. [He's an] amazing player. Tons of respect for him. For such a young player to show so much poise in important moments is impressive.
"Obviously, three sets today, but almost four hours. You just have to put your hats down to a guy like that. He plays so maturely, handles the pressure so well for a 20-year-old.
"We cannot forget how young he is. That's something that is so impressive about him."
Both players now head to New York for the US Open where Alcaraz will attempt to defend a Grand Slam for the first time, while Djokovic, bidding for a 24th Grand Slam win, returns after missing last year's tournament because of his Covid vaccination status.
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