Svitolina was cheered on by the fans despite playing an American and soaked in the applause after converting her ninth match point under the lights on a cold night in the California desert.
"I want to thank you for your support of Ukraine," she said in an on-court interview.
"It's nice to get these little wins for Ukraine."
Svitolina, seeded 23rd, will play Danielle Collins in the third round.
A Ukrainian flag flapped in the wind above Stadium Three court as Svitolina was cheered on during the match by her husband Gael Monfils, who plays American Sebastian Korda in the second round on Saturday.
Svitolina has been outspoken in support of her home country since Russia's invasion three years ago.
After US President Donald Trump publicly feuded with Ukrainian President Volodymyr Zelenskiy late last month, Washington froze military aid to Ukraine and suspended its intelligence sharing with Kyiv.
Trump said on Thursday that he would soon decide whether to revoke temporary legal status for some 240,000 Ukrainians who fled the conflict with Russia.
