The Polish world number two recovered from being broken twice in the first set against the 130th-ranked Yuan, who served for the opener before Świątek levelled and won a tight tie-break.
The Pole then built a 3-0 lead in the second set and closed out a 7-6, 6-3 victory in two hours despite Yuan's brief fightback following a medical timeout for back treatment.
"I was a bit rusty at the beginning," Świątek said in her on-court interview.
"I knew what I had to do to win. I had to become braver in my decision-making, improve my footwork," she added.
The victory keeps alive Świątek's quest for the only Grand Slam title she has yet to win.
She will face Czech player Marie Bouzková, a 6-2, 7-5 winner over Mexico's Renata Zarazua, in the second round on Thursday.
Świątek holds a 2-0 head-to-head advantage over Bouzková.
Meanwhile, Poland's Magdalena Fręch defeated Slovenia's Veronika Erjavec 6-1, 6-1, while Magda Linette recovered from losing the opening set to overcome 15th-seed Emma Navarro of the United States 3-6, 6-3, 6-3.
Grand Slam debutant Linda Klimovicova advanced when Britain's Francesca Jones retired in the second set of their first-round women's singles match.
Kamil Majchrzak completed Poland's successful day with a 7-6, 7-5, 3-6, 7-6 victory over Britain's Jacob Fearnley in the men's singles.
Hubert Hurkacz, the final Polish singles player in action, is scheduled to face Belgium's Zizou Bergs on Tuesday.
(ał/gs)
Source: PAP