Wawrinka beat the sixth-seeded Greek 7-6(6) 5-7 6-4 3-6 8-6 to set up a meeting with fellow Swiss Roger Federer who cruised past Leonardo Mayer of Argentina in straight sets.
Wawrinka, who beat Federer in straight sets en route to the 2015 French Open title, clinched victory on his second match point, with a backhand that landed on the line on a sun-soaked Court Suzanne Lenglen.
Both armed with a blistering single-handed backhand, the match between the 34-year-old Wawrinka, three-times Grand Slam winner, and the 20-year-old Tsitsipas, who had been trying to become the first Greek player to reach the Roland Garros quarter-finals, more than lived up to its billing.
Having dropped the first set on a double fault, the swashbuckling Greek won a dramatic 77-minute second on his sixth set point, having squandered a 3-1 and 5-3 lead.
The best rally of the match came during a see-saw 13-minute game at 5-4 in the second set, with Wawrinka ripping groundstrokes and smashes as Tsitsipas retrieved.
The Swiss, who had two knee surgeries in 2017, prevailed with a backhand drop shot that touched the net on the 23rd shot, to a standing ovation from the crowd.
After splitting the third and fourth sets, the 28th-seeded Wawrinka stamped his authority on his younger opponent, who had squandered three break points at 5-5 in the decider.
Serving to stay in the match at 6-7, Tsitsipas dumped a forehand into the net to hand his opponent two match points and Wawrinka took advantage to reach his first Grand Slam quarter-final since the 2017 French Open where he went on to make the final.
OTHER SUNDAY RESULTS
NADAL BREEZES INTO LAST 16
Defending champion Rafa Nadal struck 40 winners in a 6-2 6-3 6-3 victory over Argentine Juan Ignacio Londero to reach his 13th Roland Garros quarter-finals.
FEDERER MARCHES INTO PARIS QUARTERS
Third seed Roger Federer, bidding to add to his French Open title triumph in 2009, cruised into the quarter-finals with a 6-2 6-3 6-3 victory over Argentine Leonardo Mayer.
KONTA BLAZES PAST VEKIC
British number one Johanna Konta produced a sparkling display, studded with 33 winners, to claim a 6-2 6-4 victory over Croatian Donna Vekic.
Konta has reached her first Grand Slam quarter-final since her final four appearance at Wimbledon two years ago.
“To be able to win a match likes this against a really tough opponent is great,” she said.
“I felt I played well throughout the match. I feel great. To win in front of a crowd like this gives you goosebumps.”
MARTIC POWERS INTO FIRST GRAND SLAM QUARTERS
Croatia’s Petra Martic rallied from a set down to beat Estonian Kaia Kanepi 5-7 6-2 6-4 and seal progress to her maiden Grand Slam quarter-finals.
Martic, who recorded her tour-leading 15th clay court win of the season, will next face Czech teenager Marketa Vondrousova.
0945 PLAY BEGINS AT ROLAND GARROS
Vondrousova eased past Latvia’s 12th seed Anastasija Sevastova 6-2 6-0 to reach her maiden Grand Slam quarter-finals.