The German, fired up by a code violation for being coached from the stands by his father, stormed into the lead in the fourth set before securing victory in 3-1/2 hours on Court Suzanne Lenglen.
"I'm young, so I might as well stay on court and practise a little bit," Zverev said courtside. "It's the best, a five-set match.
It is the first time Zverev has reached the quarter-finals of a Grand Slam and he will face Austrian seventh seed Dominic Thiem, who beat Japan's Kei Nishikori 6-2 6-0 5-7 6-4.
Nishikori took his only break point opportunity to claim the third set but Thiem, twice Roland Garros semi-finalist, regained control in the fourth.
Former champion Novak Djokovic never looked in trouble as he booked a 12th quarter-final appearance in Paris with a 6-3 6-4 6-2 win over Spanish veteran Fernando Verdasco.
U.S. Open runner-up Madison Keys advanced to the last eight without dropping a set, defeating Romania's Mihaela Buzarnescu 6-1 6-4.
Up next for Keys is Kazakh Yulia Putintseva, who upset 26th seed Czech Barbora Strycova 6-4 6-3.
Sloane Stephens sprinted past Estonian Anett Kontaveit 6-2 6-0 to set up a potential semi-final meeting with Keys, whom she beat in last year's U.S. Open final.
Highlights from day eight (all times GMT:
1925 PLAY SUSPENDED FOR THE DAY
Play was suspended due to fading light, with Russian Daria Kasatkina leading 7-6(5) 3-3 against second-seeded Caroline Wozniacki.
1740 CECCHINATO'S DREAM RUN CONTINUES
Italian Marco Cecchinato, who had never won a Grand Slam match before this week, beat eighth-seeded Belgian David Goffin 7-5 4-6 6-0 6-3 to reach his first major quarter-final.
At world number 72, Cecchinato becomes the lowest-ranked man to reach the last eight in Paris since Latvian Ernests Gulbis (ranked 80) in 2008.
He will next face 2016 champion Novak Djokovic for a place in the final four.
"For me it is a dream," Cecchinato said. "It will be a pleasure to play against Novak."
1725 DJOKOVIC SAILS THROUGH
Former champion Novak Djokovic sprinted past Spain's Fernando Verdasco 6-3 6-4 6-2 to reach his 12th Roland Garros quarter-final.
It was the Serb's 200th Tour-level clay court match win.
1715 WILLIAMS SISTERS KNOCKED OUT
Third-seed pair of Andreja Klepac and Maria Jose Martinez Sanchez survived a mid-match blip to beat Serena and Venus Williams 6-4 6-7(4) 6-0 in the women's doubles clash.
The Williams sisters have claimed 14 Grand Slam doubles titles together since winning the French Open title in 1999, but bowed out at the last-16 stage this year.
1500 UNSEEDED PUTINTSEVA CRUISES PAST STRYCOVA
Kazakh Yulia Putintseva produced a clinical display to beat 26th seeded Czech Barbora Strycova 6-4 6-3 in exactly one hour and 30 minutes.
Putintseva matched her best Roland Garros performance by reaching the quarter-finals for the first time since 2016.
The 23-year-old will next face U.S. Open runner-up Madison Keys.
1430 STEPHENS GLIDES INTO PARIS QUARTERS
U.S. Open champion Sloane Stephens was in complete control as she broke Estonian Anett Kontaveit's serve five times on her way to a 6-2 6-0 win in just 52 minutes.
"Coming on to play on back-to-back days in a Grand Slam, I was really pleased to get out there and play some good tennis," American Stephens said.
1310 THIEM BATTLES PAST NISHIKORI
Seventh seed Dominic Thiem booked a place in the quarter-finals for a third straight year, defeating Japan's Kei Nishikori 6-2 6-0 5-7 6-4 on Court Philippe-Chatrier.
"The first two sets were amazing but then he (Nishikori) raised his levels," Thiem said.
"At the end I was nervous. It's something very special to serve for the French Open quarters.
"I love these conditions; it was the first time for me on Court Philippe-Chatrier not playing against Rafa (Nadal)... so I could enjoy the court."
Thiem will next face second seed Alexander Zverev for a spot in his third consecutive semi-final in Paris.
1250 ZVEREV SURVIVES ANOTHER MARATHON BATTLE
Second seed Alexander Zverev recovered from two-sets-to-one down for a third consecutive match to edged out Russian Karen Khachanov 4-6 7-6(4) 2-6 6-3 6-3.
The 21-year-old German has progressed to his first Grand Slam quarter-finals and became the seventh German man in the Open Era to reach the last eight at Roland Garros.
"I'm young, so I might as well stay on court and practise a little bit," Zverev said. "It's the best, a five-set match.
"I've been down 2-1 every single time but I'm unbelievably happy to be into the quarter-finals. All the hours in the gym have paid off."
1020 KEYS UNLOCKS QUARTER-FINAL SPOT
U.S. Open runner-up Madison Keys overcame Romanian Mihaela Buzarnescu 6-1 6-4 in just over an hour to reach her first quarter-finals at Roland Garros.
"Firstly, I'm really excited to be in my first quarter-finals in Paris," Keys said.
"I had a really tough opponent today but I'm really pleased with my performance. I lost to my friend Sloane in the U.S. Open final last year but I hope for big things here."
Keys, seeded 13th, has yet to drop a single set after four rounds in the tournament.
(Reporting by Hardik Vyas in Bengaluru; editing by Ed Osmond)
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