Swiatek battled to a 6-2, 7-5 victory over Irina-Camelia Begu of Romania as the 23-year-old Pole, back where she captured a fourth French Open title last month, had the advantage of playing under the roof of Court Philippe Chatrier.
"I'm happy to get through. The atmosphere was a little bit different so I was a little more nervous," she said.
"But I have the chance to work on a few things tomorrow so my level can be higher in the second round."
After two breaks in a comfortable first set, the top seed had to retrieve a break in the second set to get to 5-5 before sweeping the next two games and secured victory when Begu double-faulted.
Swiatek, who only made the second round on her Olympics debut in Tokyo three years ago, will face either Diane Parry of France or Argentina's Nadia Podoroska for a place in the last 16.
Italy's Jasmine Paolini, easily defeated by Swiatek in the French Open final, also made the second round with a 7-5, 6-3 win over Romania's Ana Bogdan.
"It's great to come back here, it's a different kind of tournament when it's the Olympics," said Paolini.
"I really love to play for my country but I feel a little more pressure."
As Swiatek and Paolini were safely in the second round, the start of play on the outside courts was delayed by more than four hours.
In all 40 matches were scheduled to be played on the uncovered courts on the opening day.
Djokovic gold-medal quest
Later Saturday, Novak Djokovic, playing in his fifth Olympic Games, starts his bid for a first gold medal to add to his 24 Grand Slams.
The 37-year-old Serb, top seed in the absence of world number one Jannik Sinner, takes on Australia's Matthew Ebden who has not played a singles match for two years and only made the draw as an alternate.
"They asked if I would be keen, and I thought, 'Hey, I would love the opportunity to play singles at the Olympics and officially retire my singles career'," said Ebden, 36.
"I've played Andy Murray many times, Rafa Nadal a few times, Roger Federer as well, but I never got to play Novak in singles, which I regretted a bit."
Victory for Djokovic would set up a potential second round blockbuster against old rival Rafael Nadal, playing at Roland Garros where he has won 14 of his 22 Grand Slams.
Nadal, a gold medallist in singles at the 2008 Beijing Olympics and doubles in Rio eight years later, is playing under an injury cloud having picked up a thigh strain.
However, he still managed to play a key role in Friday's spectacular opening ceremony where he carried the Olympic torch.
Fresh from back-to-back triumphs at the French Open and Wimbledon, Carlos Alcaraz faces Lebanon's Hady Habib, the world number 275.
Alcaraz also plays doubles later Saturday with Nadal as the Spanish pair team up for the first time.
Naomi Osaka, a four-time major winner, takes on fellow former world number one Angelique Kerber, the 36-year-old German who will retire from tennis after the Olympics.
