The American fourth seed, champion in 2023, was tested in the second set by Turkish outsider Sonmez but came through 6-3 6-4 at Flushing Meadows on Tuesday (Wednesday AEST).
"I was nervous in the tunnel, and then for some reason when the warm-up started, it went away. I think it showed the first game how I came out serving really well,'' Gauff said.
"I don't know what put my mind at ease. I think just maybe the confidence I had this week and knowing, like, you know, the opponent would have to play a really high level and me a really low level, maybe for it to not go my way.
"But Zeynep brought that high level today. Yeah, I was just happy to get out of that second set."
Gauff's victory set up a second-round match against either Paula Badosa or Daria Kasatkina.
Spaniard Bados easily took the first set 6-1 and was leading the Australian 5-4 in the second when their match was postponed because of rain.
Second seed Elena Rybakina, of Kazakhstan, advanced after beating teenage American wildcard Thea Frodin 6-3 6-2.
Frodin was Serena Williams' double in some of the tennis scenes in the Oscar-winning film about the lives of Serena and her sister Venus, King Richard, starring Will Smith.
"I actually have not met her yet, but hopefully I can get the chance soon to get the opportunity to meet her,'' Frodin said.
Switzerland's Belinda Bencic was fortunate to avoid being defaulted after whacking a ball into the crowd.
The 29-year-old, seeded 12th, swatted a ball in frustration after losing a point, then turned in horror after realising where it had flown.
She raised her arms in apology and was given a warning by the umpire, but opponent Yulia Putintseva clearly felt the punishment should have been harsher.
However, Kazakhstani Putintseva eventually won the match in three sets to send Bencic packing anyway.
After a rain-interrupted day in New York, fan favourite Alexandra Eala brightened spirits inside Louis Armstrong Stadium in the night session.
Eala, the No.17 seed from the Philippines, beat American Mary Stoiana 6-1 6-2 in front of her many Filipino fans.
There was a close call for Greek player Maria Sakkari, the No.32 seed, who lost the first set before rallying to beat American Robin Montgomery 2-6 7-5 6-2.
There was no such drama for French Open champion Mirra Andreeva, the Russian proving far too strong for Indonesia's Janice Tjen in a 6-2 6-2 thumping.
American Madison Keys, the No. 22 seed and 2025 Australian Open champion, beat Russian 19-year-old Alina Korneeva 7-5 6-4.
- with AP
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