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Nadal overcame a horrid start and a bloody nose to beat Fognini and reach the third round.
Winners of 14 Grand Slam doubles titles, including two US Opens, the Williams sisters were playing together for the first time in more than four years.
No. 12 Coco Gauff and 20th-seeded Madison Keys set up a third-round match-up.
Serena Williams stuns world number two Anett Kontaveit 7-6(4), 2-6, 6-2 and move into the third round of the US Open.
Venus lost to Alison Van Uytvanck and Emma Raducanu was defeated by Aliz Cornet. Osaka lost to Danielle Collins.
She also was simply outplayed by Cornet, a 32-year-old from France who beat past major champions Simona Halep at the Australian Open, Jelena Ostapenko at the French Open and Iga Swiatek at Wimbledon, ending the No. 1-ranked woman's 37-match winning streak at Wimbledon.
Tsitsipas was knocked out by Colombian qualifier Daniel Galan, who clinched the 6-0 6-1 3-6 7-5 win on his ninth match point.
The former champion's choice of wardrobe for Monday's first-round match landed her in a spot of bother amid windy conditions
Serena Williams advanced to the second round of the US Open on Monday with a scrappy 6-3 6-3 win over Danka Kovinic.
The World No 1 won 6-2,6-4, 6-0 at the Arthur Ashe Stadium.
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