Swiatek, who prevailed 6-3 6-2 over former champion Sloane Stephens, is the last of the top four seeds remaining at Flushing Meadows after Croatia's Petra Martic beat fourth seed Paula Badosa 6-7 (5-7) 6-1 6-2.
Serena (40) and Venus Williams (42), joining forces for their first US Open since 2014, fell 7-6 (5) 6-4 to Czech duo Lucie Hradecka and Linda Noskova, leaving Serena free to focus on what could be the final singles campaign of her career.
Belarusian Victoria Azarenka (33) triumphed 6-2 6-3 over Marta Kostyuk of Ukraine but was denied the customary handshake after a politically charged match, with the pair instead exchanging a quick tap of the racquet.
In a late night scrap on Arthur Ashe stadium, Nadal overcame a poor start and a freak injury scare to defeat Fabio Fognini 2-6 6-4 6-2 6-1, after he was struck on the nose by his racquet when it hit the court and bounced back.
Spanish teenager Carlos Alcaraz (19), quarter-finalist last year, dug deep to hold off veteran Argentine Federico Coria and clinch a 6-2 6-1 7-5 victory.
TIAFOE BEATS KUBLER IN STRAIGHT SETS
American Frances Tiafoe saw off Jason Kubler of Australia 7-6(3) 7-5 7-5 in just over three hours to reach the third round, where he will meet Argentina's Diego Schwartzman.
'SPECIAL Ks' ADVANCE AFTER COMEBACK WIN
Nick Kyrgios and Thanasi Kokkinakis, the Australian Open doubles champions, advanced to the second round after recovering from a set down to clinch a 4-6 6-3 6-4 victory over Hugo Gaston and Lorenzo Musetti.
IVASHKA UPSETS HURKACZ
Belarusian Ilya Ivashka stunned eighth seed Hubert Hurkacz 6-4 4-6 7-6(5) 6-3 to reach the third round, where he will meet Lorenzo Musetti of Italy.
SHAPOVALOV PROGRESSES
Canadian 19th seed Denis Shapovalov overcame a slight wobble in the second set to get the better of Spain's Roberto Carballes Baena 6-4 4-6 6-3 6-2. He next faces Andrey Rublev.
COLLINS TOO GOOD FOR BUCSA
Australian Open runner-up Danielle Collins eased into the third round with a commanding 6-2 7-5 win over Spain's Cristina Bucsa. Up next for the American is Frenchwoman Alize Cornet.
SABALENKA FIGHTS BACK TO OUST KANEPI
Belarusian sixth seed Aryna Sabalenka, a semi-finalist last year, roared back from 5-1 down in the second set and saved two match points along the way to overcome Estonian veteran Kaia Kanepi 2-6 7-6(8) 6-4.
Former finalist Karolina Pliskova also booked her place in the third round with a comfortable 6-3 6-2 win over fellow Czech Marie Bouzkova.
NORRIE THROUGH IN STRAIGHT SETS
British number one Cameron Norrie was barely troubled in a convincing 6-4 6-4 7-6(4) win over unseeded Portuguese Joao Sousa.
RUBLEV BLASTS PAST KWON
Andrey Rublev, a two-times quarter-finalist in New York, swept aside South Korea's Kwon Soon-woo 6-3 6-0 6-4 to move into the third round.
Twice major winner Garbine Muguruza also progressed after a dominant 6-0 6-4 victory over Linda Fruhvirtova.
SINNER SAILS INTO THIRD ROUND
Italian 11th seed Jannik Sinner survived a tough second set to beat American Christopher Eubanks 6-4 7-6(8) 6-2. He next faces Brandon Nakashima.
MARTIC STUNS BADOSA
Spanish fourth seed Paula Badosa became the latest high-profile casualty in the women's draw, going down 6-7(5) 6-1 6-2 to Croatia's Petra Martic.
BENCIC SINKS CIRSTEA
Olympic champion Belinda Bencic fought back from a set down to see off Sorana Cirstea of Romania 3-6 7-5 6-2 and move into the third round.
In the men's draw, former champion Marin Cilic made light work of Spaniard Albert Ramos-Vinolas, winning 6-3 7-6(4) 6-3 to advance. Up next for the Croatian 15th seed is Dan Evans.
PEGULA ADVANCES TO THIRD ROUND
American eighth seed Jessica Pegula booked her place in the third round after overcoming Belarusian Aliaksandra Sasnovich 6-4 6-4.
KVITOVA GETS WALKOVER AFTER KALININA WITHDRAWS
Czech 21st seed Petra Kvitova, who has twice reached the quarter-finals at the US Open, received a walkover into the third round after her Ukrainian opponent Anhelina Kalinina withdrew due to illness.