The 25-year-old had never won a Grand Slam match before this tournament, but showed some good form when winning an ATP clay-court title in Budapest as a lucky loser earlier this year.
The Bulgarian, a two-time Grand Slam semi-finalist, has still never reached the second week at Roland Garros in eight attempts.
His opponent, however, was the stronger in an epic fourth-set tiebreak, prevailing on his fourth match point.
Djokovic won the tournament in 2011, 2014 and 2015 but has failed to make it beyond the quarter-finals in the past two years.
However, Verdasco held his nerve under intense pressure, saving all four set points at 5-6, one of them with a second-serve ace that flew down the middle.
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However, she recently tweeted that she was "working hard" and "feeling great", but some believe that the FLOTUS didn't write it. Four days later, the White House announced that she had been hospitalized to treat a kidney condition described only as benign.
As his backhand return on his fourth match point landed beyond the outstretched arm of Djokovic, the tears flowed. "So after Roland Garros, I need some rest and to realize the moment, and we will see my life", the world number 72, the lowest-ranked player to reach this stage at Roland Garros since Andrei Medvedev in 1999, told reporters.
'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.
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. What can I say? "It's fantastic", he said.
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 20th seed reached the last 16 - where he will face Fernando Verdasco - at Roland Garros for the 12th time in his career after triumphing 6-4 6-7 (6-8) 7-6 (7-4) 6-2 on Suzanne Lenglen.
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The former Democratic president added, "And most people I know believe that the press would have been that hard, or harder". In 2018, Ms Lewinsky discussed the movement in a Vanity Fair essay .
Like her compatriot and good friend Keys, US Open champion Stephens is in the quarter-finals for the first time and hoping to set up a repeat of last year's final in NY.
"I definitely thought about it, but I didn't want to pull out for the first time of my career in a Grand Slam quarter-final", said Zverev.
"It was the best moment of my life", Cecchinato said. "Now I beat David Goffin, (who) is one of the best players in the world". "It's incredible", said the 25-year-old Cecchinato. "So it's going to be an awesome challenge for me".
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.
"How? I mean, we'll figure [it out]".
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Mexico said it would levy tariffs on American farm products, while Canada zeroed in on the same metals that Trump has targeted. The EU, Canada and Mexico also threatened stiff retaliatory tariffs as they pushed back against Trump's moves.
Stephens, the 10th seed, had previously made the fourth round for four successive seasons and will next play either Australian Open champion and second seed Caroline Wozniacki or Russia's Daria Kasatkina.