The 42-year-old beat Volynets, who is half her age, 7-6 (7-4) 6-2 and will play China‘s Lin Zhu next.
Former world number one Williams made her Tour debut in October 1994.
She entered four tournaments as a wild card in 2021, the only year since 1994 in which she did not win a singles match.
Her victory over Volynets, who turned 21 on New Year’s Eve, was interrupted by rain during the first set tie-break, led by her opponent.
But Volynets faded, as the match dragged on into the night, with Williams notching her first singles win since Wimbledon in 2021.
“It’s hard when you’re excited and all of a sudden you have to stop,” said Williams, who is now in the ranked 1,003 in the world.
“I was so angry on the rain break.
My coach could barely speak to me, so I’m going to try to improve my attitude.
Williams was given a wild card to compete at the Australian Open, 25 years after first playing in the tournament.
Williams, a seven-time Grand Slam singles champion, has never won the singles title in Melbourne, losing to her now-retired younger sister Serena in the 2003 and 2017 finals.
The Australian Open begins on January 16 .