The Belarusian was watched by a very sparse crowd in the giant Arthur Ashe Stadium but did not let that bother her as she raced into a 5-0 lead against Larsson and did not look back, winning 6-1 6-3.
Title favourite Williams, seeded 28th after injury and health problems, opens her campaign in the night session against Bojana Jovanovski, and she will meet Azarenka at the weekend provided both make it to the third round.
The draw is rough luck on the 22-year-old, who made it to the semi-finals of a grand slam for the first time at Wimbledon but lost convincingly to Williams in Toronto earlier this month.
Another Wimbledon semi-finalist, Sabine Lisicki, is also through to the second round where she will meet Venus Williams, who is unseeded at the US Open for the first time since 1997.
Lisicki's 6-3 6-2 win over Alona Bondarenko was a particularly good effort considering the 22nd seed did not make it to New York until.
The German won the WTA Tour title in Dallas on Saturday, her second of the season, but Hurricane Irene made travel to and from New York extremely difficult over the weekend.
Seeds Yanina Wickmayer, Flavia Pennetta and Kaia Kanepi all made it safely through their first-round encounters.
Lisicki did not watch Venus' win over Vesna Dolonts, who had an even more stressful journey from Russia and arrived only four hours before her match, because she was having her only practice after finally making it to New York at 4pm.
The German said: "I just had one practice and warm-up this morning. That was it. It's not easy but I'm a tough girl. That's fine."
Lisicki has played Venus twice before, winning in Charleston in 2009 and losing in Dubai last year, and she was philosophical about having to play the two-time former champion at such an early stage of the tournament.
She said: "Twice we've had a very, very good match. I think it really depends who plays better that given day and that person will win. That's how it was the last two times.
"I just want to fight and play my best and see what happens. The Williams sisters always play well at the grand slams, and you can never underestimate them.
"So it's for sure going to be a tough match. A very good challenge. But, if you want to succeed, you have to beat the best."
Azarenka, who has recovered fully from a wrist problem that affected her in Toronto, was content with her performance.
The fourth seed said: "There are always things that could have been done better, but I think I had a pretty solid start. I kept control of the match. It's important to have that control in the opening rounds."