The fourth-seeded Fritz served four aces and won 88% of his first-serve points against the 30th-seeded Baez from Argentina.
Rune saved a match point and rallied from 2-5 down to win the second-set tiebreaker. But Wawrinka recovered in the third to send his 19-year-old opponent packing.
“The last 45 minutes was tough,” Wawrinka said. “I had the match in control and was playing well. I was putting a lot of pressure, I should have won the match way before.”
Wawrinka reached the fourth round in the Southern California desert for the first time since 2017, when he lost in the final. He turns 38 next month.
“All my career I know only one way,” Wawrinka said. “To keep fighting, working, pushing myself and stay positive. I am super happy with the win.”