Djokovic, seeded 20th, won 6-3 6-4 6-2 against 30th seed Verdasco.
Djokovic eased past Spanish veteran Verdasco 6-3, 6-4, 6-2 and will play in the last eight at Roland Garros for a record 12th time.
The Serbian, who has struggled with injury for much of the previous year, was asked when he planned to make his first grass-court appearance.
With world number two Caroline Wozniacki and defending champion Jelena Ostapenko already eliminated, the bottom half of the women's draw has been throw wide open.
No. 13 Madison Keys also plays on Tuesday against Yulia Putintseva.
US Open champion Sloane Stephens cruised past Estonian Anett Kontaveit 6-0, 6-2 to book a last-eight spot for the first time.
New Facebook privacy furore: What's at stake?
Apple , Microsoft , Samsung and Amazon were among the companies who signed up to data sharing agreements using the APIs. Zuckerberg was adamant before Congress that Facebook is seriously committed to users' privacy.
Read on as we provide highlights and reaction from a whirlwind day of quarter-final action at Roland-Garros.
Playing at his lowest ranking for nearly 12 years after months of struggling with an elbow injury and indifferent form, Djokovic has made the last eight by dropping just one set.
'I think that's going to be the No. 1 priority for me now to kind of step out from the tennis for a little bit, try to watch some matches and sort of try to progress somehow and just get better.
The 25-year-old from Sicily was suspended for 18 months and fined 40,000 euros (about $45,000) by his national federation in July 2016, accused of losing on goal at a lower-tier Challenger event in Morocco a year earlier.
Spaniard Verdasco had earlier demonstrated his class on clay with a straightforward win over disillusioned fourth seed Grigor Dimitrov.
Draymond Green Says He’s Willing to Fight Tristan Thompson Off the Court
Calderon's roundabout journey to his first finals began in Spain, where he grew up watching his father play for Doncel La Serena. Coach Steve Kerr had said Thompson, who suffered a sprained left ankle in Game 1 on Thursday, would be evaluated after warmups.
"It's a pleasure playing against Novak in quarter-final at Roland Garros", said the Italian who told reporters that he would not discuss the scandal that could have ended his career.
Verdasco wavered for the only time in the match when he blazed a forehand long to allow Dimitrov to break back in the third set having established a 4-2 lead.
The 25-year-old, who was ranked 957 just six weeks before her maiden Grand Slam triumph last year after a series of injury problems, is now into the last four of a major for the third time.
The 31-year-old Serb said he did not "want to talk about tennis right now".
In a tense encounter, both men were slapped with code violations for coaching before Cecchinato gallantly saved three set points.
USA may open door to more temporary immigrant workers, Trump aide says
The May jobs report was in fact a good one for a president who likes to brag about the strong jobs market. After Trump's tweet, the two-year Treasury yield climbed one basis point, while the dollar stayed flat.