"I'm happy to get through. This was an ugly match for me,'' she said.
All of the top 10-seeded women have reached the third round at the Australian Open for the first time since 2007.
It's the first time it's happened at any Grand Slam tournament since Wimbledon in 2009.
Said Svitolina, who reached a pair of Grand Slam semifinals last season: "Happy I could finish in two sets and we can all go to bed now.''
Heavy morning rain showers delayed play on the outside courts for at least 90 minutes and up to three hours.
Danielle Collins is out in the second round a year after making her Grand Slam breakthrough with a run to the semifinals at Melbourne Park.
The 26th-seeded American bowed out Thursday with a 6-4, 2-6, 7-5 loss to Yulia Putintseva of Kazakhstan.
Collins had never won a match in five tries at Grand Slam tournaments until her surge to the final four in Australia in 2019. She won five matches in a row to get that far, including upsets of three seeded opponents.
Muguruza had to wait for the stadium roof to be closed in the third set of her victory and said remaining patient was crucial in the tough win.
"I'm super happy to get the match out of the way. I didn't know the courts were, like, wet and dirty and all that,'' Bencic said. "But for sure, it was difficult today. It was sometimes sunny and then suddenly was almost raining, then windy. Yeah, you just kind of have to accept it and go with it and try your best.''
Two years since she was healthy enough to participate in a Grand Slam tournament -- and four arm operations later -- 20-year-old Californian CiCi Bellis beat a seeded opponent to reach the third round. Bellis eliminated No. 20 Karolina Muchova of the Czech Republic 6-4, 6-4.
Bellis was a teen prodigy, winning a US Open match at age 15 and reaching No. 35 in the rankings at 17. Her time away from the tour due to injuries has her at No. 600, but she was able to get into the draw at Melbourne Park via the protected ranking rule.
Halep ended 2019 ranked fourth, her sixth consecutive year-end top-five finish. She will next play Putintseva.
At Melbourne Park in 2016, Kerber became the first German woman since Steffi Graf in 1999 to win a Grand Slam singles title. She won the US Open later that year and Wimbledon in 2018.
Organizers were again forced to postpone matches because of the weather, rescheduling seven women's and six men's first-round doubles matches to Friday.
The Associated Press contributed to this report.