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Kazakh Yulia Putintseva who battled back superbly won the third-round clash 3-6 6-1 6-2 against Swiatek.
Djokovic defeated 277th-ranked Jacob Fearnley while top seed Swiatek, fresh from her Roland Garros success, beat Croatian Petra Martic 6-4 6-3.
Men's defending champion Carlos Alcaraz, seeded three, will start against Estonian qualifier Mark Lajal and finds himself in the same half of the draw as top seed Jannik Sinner.
The World No 1 defeated the Italian 6-2, 6-1.
Swiatek defeated Coco Gauff 6-2, 6-4 while Paolini downed Andreeva 6-3, 6-1.
Second seed Sinner kept his hopes of taking Djokovic's top ranking alive with a battling 2-6 6-3 6-2 6-1 win over local favourite Corentin Moutet.
Andrey Rublev became the highest men's seed to crash out at Roland Garros.
The defending champion Swiatek saved a match point in a high-voltage second-round match.
The title was the 21st of Swiatek's career as the world No.1 extended her unbeaten run to 12 matches on clay to sound out a warning to her rivals ahead of the French Open.
Victory improved Swiatek's record over her Belarusian rival to 7-3 as the three-time French Open champion bagged the only major claycourt title that was missing on her glittering resume.