Alexander Zverev said he had begun to feel his stomach trouble coming on in the second set, and he was seen briefly by a doctor who gave him some tablets in the third.
The 6-7 (6), 2-6, 6-3, 6-2, 6-4 defeat was the first major final for Tsitsipas and 29th for Djokovic, who also won the French Open in 2016, along with nine titles at the Australian Open, five at Wimbledon and three at the US Open.
Nadal, aiming for a record-extending 10th Rome title, will next face big-serving American Reilly Opelka, who reached his first Masters semi-finals
The 20-time Grand Slam winner, Rafael Nadal, next plays the 13th seed Canadian Denis Shapovalov for a place in the quarter-final.
Spain's Rafael Nadal advanced to the semifinals at the ATP Finals for the first time in five years after knocking out the defending champion Stefanos Tsitsipas.
World number one Novak Djokovic, the 2016 champion at Roland Garros, reached his fifth final in Paris, defeating fifth seed Tsitsipas 6-3, 6-2, 5-7, 4-6, 6-1.
Greece's Stefanos Tsitsipas defeats Austria's Dominic Thiem 6-7 (6/8), 6-2, 7-6 (7/4) during the men's singles final match in London on day eight of the tournament.
The 32-year-old Serbian tennis star avenged his loss to the 21-year-old Greek 3 weeks earlier in the Shanghai Masters, with a 58-minute victory.
US Open runner-up Medvedev, chasing his fourth ATP Tour title in a breakthrough 2019 season, will meet Germany's Alexander Zverev in Sunday's final.
Tsitsipas had never won a set in his previous three meetings against claycourt specialist and 11-times French Open champion Nadal, but the Greek drew first blood by saving a break point to hold in the 10th game and take the first set.
Rafael Nadal showed no signs of rust as he raced into a 5-0 lead against fellow Spaniard Roberto Bautista Agut in the first set on Court Rainier III, defending his Monte Carlo Masters title with 6-1, 6-1.
Swiss player Roger Federer overcomes Stefanos Tsitsipas 6-4 6-4 in Dubai final.
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