In other women's games on Monday, defending champion Swiatek needed less than hour to defeat Sweden's Rebecca Peterson 6-0, 6-1
The 22-year-old world No 1 from Poland overwhelmed her opponent with five breaks of serve and 20 winners to complete victory in just 58 minutes on the Arthur Ashe Stadium.
"I wanted to start the tournament with everything I was focusing on. I was happy to play a great game, with all the pressure and expectations I was happy on the court," said Swiatek after her win.
Swiatek, targeting her fifth Grand Slam title, will face Daria Saville of Australia in round two.
While Swiatek advanced safely, there was no such luck for Greece's eighth seed Maria Sakkari, who became the highest ranked early casualty, beaten 6-4, 6-4 by Spain's Rebeka Masarova, ranked 71st in the world.
Kazakhstan's fourth seed Elena Rybakina was an easy winner over Ukraine's Marta Kostyuk, cruising through to the second round with a 6-2, 6-1 victory.
There was also a win for former world No 1 Caroline Wozniacki, making her first Grand Slam appearance in over three years.
The 33-year-old Danish mother-of-two, who came out of retirement earlier this month, defeated Russian qualifier Tatiana Prozorova 6-3, 6-2.
Back in the men's draw meanwhile, Djokovic's route to a potential final meeting with Alcaraz was made easier as Danish fourth seed Holger Rune was stunned in four sets by Spanish journeyman Roberto Carballes Baena, a clay-court specialist ranked 63rd in the world.
Carballes Baena punished an error-strewn display by Rune to win 6-3, 4-6, 6-3, 6-2.
Last year's US Open runner-up, Norway's fifth seed Casper Ruud, reached the second round with a battling 7-6 (7/5), 3-6, 6-4, 7-6 (7/5) defeat of Emilio Nava of the United States.
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