The 33-year-old Swiss wild card beat the Bosnian, who was seeded sixth, 6-3, 6-4 in one hour and 11 minutes to level their head-to-head record at one all.
“The level of my play was good today and the win makes me feel positive,” Wawrinka said.
In the last four, Wawrinka will face Slovak Martin Klizan, the 2012 champion, who battled back from a set down to beat seventh-seeded Denis Shapovalov of Canada 3-6, 7-5, 6-3.
“I expect an interesting match with Klizan in the semis,” Wawrinka said. “We ve played a couple of times in the past and it was always really tough.”
Wawrinka, the former world number three who is down at 88 after struggling to find his best form following two operations on his left knee, looked nervous at the start dropping his serve early in the opening set.
He responded by breaking back twice to take the opener in 32 minutes.
In the second, Wawrinka broke early and experienced few troubles.
French Open runner-up Dominic Thiem, the top seed in St Petersburg, also booked a place in the event s semis beating the local crowd s favourite Daniil Medvedev, seeded eighth here, 6-2, 3-6, 7-6 (7/2) in a hard-fought first meeting between the two.
The 25-year-old Austrian, seeking his third title of the year, looked in complete command from the start breaking Medvedev s serve twice to take the opening set in 30 minutes.
Medvedev fought back and broke halfway into the second set to pull the scores level after one hour nine minutes on court.
The opponents traded breaks in the deciding set, which ended in a tie-break which Thiem dominated.
“It was a great match from the beginning to the end,” Thiem said. “The first set was easier for me, the second easier for him. The tie-break in the deciding set is always a coin flip. I m very lucky that I won today.”
In the semis, Thiem will take on fifth seeded Spaniard Roberto Bautista Agut, who saw off Italian third seed Marco Cecchinato 7-6 (7/0), 6-3, also in the pair s first ever meeting.