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The 2017 Roland Garros champion next faces American sixth seed Coco Gauff, who beat wildcard Caroline Wozniacki in three sets earlier in the day.
The British charge at the US Open will carry on without 2012 champion Andy Murray, who was outclassed by Bulgarian Grigor Dimitrov.
Samsonova had to play the match about two hours after upsetting third seed Elena Rybakina in a rain-postponed semifinal clash meant to be played on Saturday.
Weronika Falkowska and Katarzyna Piter lost 6-4, 6-4 to Heather Watson and Yanina Wickmayer in the doubles final.
The World No 1 defeated his childhood friend Holger Rune of Denmark in the quarterfinals.
The Serb won the quarterfinals, 4-6 6-1 6-4 6-3, to stay on track for his fifth straight title.
After almost three hours of nerve-jangling drama, Svitolina gave her war-ravaged homeland something to cheer about when she pulled off the biggest upset of this year's championships.
Protests by Just Stop Oil campaigners interrupted play twice during matches on Court 18.
The world number one defeated Zhu Lin 6-1, 6-3 to enter the second round.
Friday's draw at the All England Club pitched women's top seed Iga Swiatek against China's 33rd-ranked Zhu Lin, a potentially tricky first hurdle for the Pole.