"He served for the match and I had to save lots of break points in the fourth and fifth sets".
However, he threw away a 5-2 lead in the fourth set and ended up losing a dramatic tie-break.
The Italian, who has one career title to Djokovic's 68, had never won a match at the majors before this year's French Open.
Elsewhere Alexander Zverev fought a five-set thriller against Bosnian Damir Dzumhur on the Philippe Chatrier court, winning 2-6 6-3 6-4 6-7 (3-7) 5-7 in a match that lasted three hours and 54 minutes.
'But even so, I still had some time to kind of get into a rhythm, get into a different match, and just didn't happen.
"It was an wonderful, high level match", said Zverev.More news: Bryson DeChambeau wins the Memorial Tournament in playoff
Seeded 20th after a well-documented slump, Djokovic blunted Verdasco's explosive firepower with his signature counter-punching to book an unlikely last-eight date with Marco Cecchinato. "So it's going to be an wonderful challenge for me".
US Open champion Stephens brushed aside Russian Daria Kasatkina to set up a repeat of last year's Flushing Meadows final with fellow American and good friend Keys.
After a good start where he reached the semi-finals in Monte-Carlo and the last-eight in Barcelona, he lost back-to-back matches in Madrid and Rome.
The Serbian has now made the second week at Roland Garros for the last nine years in a row.
Dominic Thiem took advantage of Alexander Zverev's depleted stamina to become the first qualifier for the men's semi-finals, beating the German 6-4, 6-2, 6-1 as the difference in court time at the competition told the story.
The 31-year-old Serb said he did not "want to talk about tennis right now".More news: Holtby, Capitals deliver resilient Game 2 performance
"For me, this is the best moment of my life".
2016 champion Djokovic, in a 12th Roland Garros quarter-final and 40th at the majors, is bidding to reach his 32nd Grand Slam semi-final when he faces world number 72 Cecchinato.
Yet Djokovic won the third set at a canter, and a suddenly frustrated Cecchinato was hit with a point penalty for unsportsmanlike conduct. "Now I beat David Goffin, (who) is one of the best players in the world".
Will Halep make it third time lucky?
Keys struck 30 winners past a battling Putintseva in an impressive display as she continued her recent revival on clay, having never reached the quarter-finals in Paris before this week.
"Honestly, the (US) Open feels like it was 12 years ago at this point", said Keys, who is yet to drop a set in the tournament.More news: LeBron James' eye remains bloodshot in Game 2 of 2018 NBA Finals
"Then once you get to the second week, every match obviously has more nerves, and there's more on the line and all of that".
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