Svitolina, seeking her sixth title of the year and first Grand Slam crown, held off 42nd-ranked Czech Katerina Siniakova 6-0, 6-7 (5/7), 6-3 while Thiem finished off Australian Alex de Minaur 6-4, 6-1, 6-1 under sunny skies.
There were 87 matches scheduled Wednesday in the year's final Grand Slam event after rain washed out most of Tuesday's agenda, 11 of them halted first-round affairs.
Former world number one Maria Sharapova, who in her Grand Slam return from a 15-month doping ban upset second-ranked Simona Halep, was set to face Hungary's 59th-ranked Timea Babos in a later second-round match at Arthur Ashe Stadium.
She played only one hardcourt tuneup match due to a forearm injury but powered past one of seven contenders for the world number one ranking still in the field.
Another of those is Svitolina, who dropped six of nine tie-breaker points when her match resumed, but survived a third set, breaking for a 4-2 edge and serving out for the victory.
"It was a little bit unlucky to stop because I was playing good," Svitolina said. "Today was a little bit of a mess in my head. It was tough to keep my focus. I'm happy I could win the third set and play really good tennis.
Thiem, a semi-finalist at the past two French Opens, was working to adapt his skill on the Paris red clay to maximum impact on the New York hardcourts.
"I'm pretty pleased with the performance," Thiem said.
"I'm trying to improve my hardcourt game. I cannot play my clay game here. I can't stay close to the baseline."
Canadian 18-year-old Denis Shapovalov, who faces French eighth seed Jo-Wilfried Tsonga later, can become the youngest man in the third round of a Grand Slam since Australia's Bernard Tomic at the 2011 Australian Open and in the US Open since American Donald Young in 2007.
Wimbledon champion Garbine Muguruza, the third seed from Spain, can reach the US Open third round for the first time by beating China's 92nd-ranked Duan Ying-Ying.
And seven-time Grand Slam champion Venus Williams, this year's Wimbledon and Australian Open runner-up, meets France's Oceane Dodin. The US ninth seed is the oldest woman in the field at 37.
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