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The world number three secured a 6-1, 4-6, 6-4, 6-7 (7/5), 6-1 win in a four-hour and 16-minute final on Court Philippe Chatrier, ending his long wait for a maiden Grand Slam title.
Zverev became the first German to win a Grand Slam since Boris Becker at the 1996 Australian Open, marking his fourth Grand Slam final appearance and second at Roland Garros.
Italian Flavio Cobolli, ranked 14th, was aiming to become the first Italian in 50 years to win the French Open but came up short in a closely contested final.