Six-time Grand Slam champion Swiatek was in all sorts of trouble, broken three times by the hard-hitting Yuan, whose game was levels above her 130th ranking.
But the Pole's backhand proved decisive as she smacked a slew of winners in the clutch points to keep Yuan at bay.
"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."
It took some backhand magic for Swiatek to deny Yuan from taking the first set as she served at 5-4.
The Pole won the tiebreak off the same flank, planting her feet on the baseline to unleash a furious inside-out winner before racing to a 3-0 lead in the second set.
With the match slipping away, Yuan called a medical timeout, complaining of tightness in her thigh area, and lay down on the court to let trainers go to work.
When she got to her feet, she promptly broke Swiatek and rallied to 3-2, triggering calls of "Jiayou!" (Come on!) from Chinese fans in the terraces.
Yuan scrapped hard to the finish, saving a match point on serve at 5-3, but there was no denying Swiatek, who closed it out with a sizzling backhand return down the line two points later.
Swiatek will meet Czech Marie Bouzkova for a place in the third round as she continues her quest for the career Grand Slam, having won crowns at the other three majors.