The six-time Grand Slam champion prevailed 7-6(5), 6-3, but was made to work far harder than the scoreline suggested by the world No 130, whose level was well above her ranking.
Swiatek was broken three times during the contest and spent long periods under pressure, particularly in a tight opening set.
Yuan served for the first set at 5-4 and threatened a major upset, only for Swiatek’s backhand to prove decisive. The Pole struck a series of backhand winners at crucial moments to deny Yuan, before sealing the tiebreak with a superb inside-out backhand winner from the baseline.
Swiatek appeared to take control early in the second set, racing into a 3-0 lead. However, the momentum shifted when Yuan took a medical timeout, complaining of tightness in her thigh.
Trainers treated her on court as she lay down, and on resumption Yuan immediately broke Swiatek and rallied back to 3-2.
Yuan later saved a match point on serve at 5-3, but Swiatek regrouped and closed out the contest two points later with a backhand return down the line.
“I was a bit rusty at the beginning, didn’t really start well,” Swiatek said on court.
“But I knew if I put in the hard work, I will play better so that’s what I tried to do.
“I’m happy that it worked, many ups and downs. There are things to work on.”
Swiatek will face Czech player Marie Bouzkova next for a place in the third round as she continues her pursuit of a career Grand Slam, having already won the other three major titles.
