The 2014 winner of the Grand Slam competition was leading 7-6 (7/2), 6-3 when Tursunov called it quits at the start of the third set.
Tursunov had called the trainer midway through the second set to treat an apparent hip problem.
The Russian told the chair umpire he was unable to continue any further.
"That's not the way you want to win, Wawrinka said.
"Never happy with an injury of an opponent, but that's part of the game," he said.
But the Lausanne native aded: "I'm happy to get through the first one.
"I don't think he changed his game much because of the injury," the fourth seed said.
"I think after a little bit he couldn't really push more, so that's why he stop."
Wawrinka, who beat Novak Djokovic and Rafael Nadal on the way to winning the Australian Open two years ago, will face veteran Czech qualifier Radek Stepanek in the second round.