Zverev had started brightly, taking the first set 6-2 but the 26th seed rallied to get back into the match and move in front at two sets to one up.
Thiem, the only man to beat Rafael Nadal on clay this year, held off a short-lived Nishikori revival on Court Philippe Chatrier to set up the meeting with Zverev.
It is the first time the German has reached the last eight of a Grand Slam - and BBC tennis correspondent Russell Fuller claimed the 21-year-old will see it as a pivotal moment in his career.
The world number 72, who had never won a Grand Slam match before this week, stunned Belgian eighth seed David Goffin 7-5, 4-6, 6-0, 6-3.
"I think psychologically that is a big, big breakthrough for Zverev today".
"I'm expecting another five-set match".
Zverev produced some stunning tennis in the closing stages of a dramatic fourth-round victory and will next face Dominic Thiem who reached a third successive quarter-final in Paris by seeing off Japan's Kei Nishikori 6-2, 6-0, 5-7, 6-4.
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"I and my brother (Mischa) are guys that spend three, four hours a day in the gym, lifting heavy weights, on the treadmill".
Ottawa's Gabriela Dabrowski and Croatia's Mate Pavic cruised into the quarter-finals of the mixed doubles tournament with a 6-2, 6-3 win over France's Eduoard Roger-Vasselin and Andrea Sestini Hlavackova of the Czech Republic on Sunday.
It wasn't easy but Alexander Zverev is into the fourth round of Roland Garros.
The Serbian has now made the second week at Roland Garros for the last nine years in a row. "It's the best, a five-set match".
Thiem, a semi-finalist at Roland Garros for the past two years, has won eight of his 10 ATP titles on clay and dominated on his favoured surface. The match kicked off with two breaks of serve and they found the range on serve to bring the next six games home easily before Sascha earned three break points in game nine.
The third-seeded pairing of Andreja Klepac and Maria Jose Martinez Sanchez beat the Americans 6-4, 6-7 (4), 6-0 to advance to the quarter-finals.
Later on Lenglen, Australian Open champion Caroline Wozniacki - seeded second - had a tough-looking fourth-round match against Daria Kasatkina.