Verdasco too failed to break his serve initially before regaining composure and winning four games in a row to take the set 6-2.
Djokovic made his feelings on the defeat clear in the post-match press conference, giving short answers primarily in his native Serbian tongue.
Paris: World number 72 Marco Cecchinato became the first Italian man in 40 years to reach a Grand Slam semi-final on Tuesday with a breathtaking 6-3, 7-6 (4), 1-6, 7-6 (11) victory over 12-time Major victor Novak Djokovic.
"In the most important moments I managed to play an extra shot in the court, made him run and made him uncomfortable". What can I say?
German second seed Alexander Zverev and Novak Djokovic edged closer to a possible French Open semi-final clash of generations, with the youngster edging out Karen Khachanov and the former champion seeing off Fernando Verdasco in the last 16 on Sunday.
Seventh seed Dominic Thiem muscled his way into the fourth round of the French Open, putting out Italian claycourt specialist Marco Berrettini 6-3 6-7 (5) 6-3 6-2 to set up a tie with Nishikori.
"I don't know. I don't know what I'm going to do".
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.
The clay courts of Roland Garros host the second Grand Slam of the season.
"For me, this is the best moment of my life". In the final tiebreak, Djokovic had a set point three times, but could not deliver.
Keys, the US Open runner-up to Stephens previous year, will face Kazakhstan's Yulia Putintseva for a place in the semi-finals, after the world number 98 saw off Barbora Strycova 6-4, 6-3.
The 23-year-old Svitolina, who trains at Roland Garros and was a quarter-finalist in 2015 and 2017, had been tipped as one of the potential winners for the second year in a row after winning the Italian Open, but she never found her groove.
He arrived at Roland Garros with a 0-4 mark in the majors, and dropped the first two sets in the first round before coming all the way back to win 10-8 in the fifth.
"I'm a better player in general, for sure".
Zverev, considered among a select few capable of dethroning defending champion and French Open king Rafael Nadal, had to do it the hard way as he was on the right side of a 6-2, 3-6, 4-6, 7-6, 7-5 scoreline.
Keys had always struggled on clay before this year, but has now reached the last eight at all four Grand Slams after impressively cruising through the draw without dropping a set.
"To be able to win that first set and get ahead was crucial. Today, (that) I won this match is really incredible, reaching the semifinals I'm happy for having written a place in history".