The 2016 Paris champion is still not at his dynamic best following a long absence due to an elbow problem, but he did not have to be to get past the 30th seed.
Djokovic took the first set, a slugfest of deuces which lasted almost an hour, with a solitary break of serve.
He repeated the trick in the second, taking eight points in a row on the way, but not without complaining to the umpire about the slippery court.
Meanwhile, Verdasco had slipped off a trainer to reveal toes heavily covered in plasters while receiving treatment for a blister on the sole of his foot.
The Spaniard grimaced his way through the third set as Djokovic broke twice to run out 6-3 6-4 6-2 winner.
Alexander Zverev has become marathon man of Roland Garros after coming through another five-setter to book his place in the quarter-final.
The 21-year-old German, seen along with Djokovic as one of the few obstacles to Rafael Nadal claiming an 11th title, was 2-1 down in sets for the third straight match, this time against Karen Khachanov.
But second seed Zverev roared back to oust the 6ft 6ins Russian, ranked 38 in the world, 4-6 7-6 (7/4) 2-6 6-3 6-3.
Despite his lofty seeding this is the first time Zverev had reached the last eight of a grand slam.
He has now played 18 sets to Nadal’s nine, with the Spaniard facing unseeded German Max Marterer on Monday in his fourth-round tie.
But Zverev insists he is more than happy to keep going the distance.
He said: “It’s about trying to find a way and trying to win. That’s what I am able to do in the last few matches, and I’m very happy about that.
“I’m very happy about being in the quarter-finals here going the hard way, going the long distance every single time and, you know, showing myself, showing everybody that I can play for as long as I need to.
“I’m in the quarter-finals of a grand slam. If you’re mentally fatigued, then something is wrong with you”
A 12th Roland-Garros quarterfinal for @DjokerNole.
— Roland-Garros (@rolandgarros) June 3, 2018
Provided by Press Association Sport
Dominic Thiem said on Sunday that since he will soon turn 25, it is time for him to finally taste Grand Slam success.
Thiem made it to the the French Open quarter-finals for the third straight year on Sunday after defeating Japan’s Kei Nishikori 6-2, 6-0, 5-7, 6-4.
In the last two years, the Austrian has not managed to go beyond the semi-finals at Roland Garros.
At the Australian and US Open and Wimbledon, Thiem has never gone beyond the fourth round.
“I think for me it’s time to move on to make a great step, because I’m turning 25 (in September). I’m not that young anymore,” Thiem was quoted as saying by AFP.
Thiem, ranked at eight in the world, used to carry tree trunks to improve his body mass and strength.
In previous editions, it was Novak Djokovic and Rafael Nadal who defeated him at the French Open semi-finals in 2016 and 2017.
Thiem said trying to win a major while Roger Federer, Nadal, Djokovic and Andy Murray are playing has been difficult.
“At the Grand Slams, there are the ‘Big Four’. And the last ten years you needed to beat at least two of them in a Grand Slam to win, and not many players did it,” said Thiem.
“Maybe it gets a little bit easier in the future when they are not that young anymore or when some of them stop.”
Meanwhile, German second seed Alexander Zverev secured a dramatic 4-6, 7-6 (7/4), 2-6, 6-3, 6-3 fourth-round win over Karen Khachanov and will face Thiem in the last eight.
The former world number one cut a frustrated figure for long periods of his clash with Spain’s Roberto Bautista Agut, the 13th seed.
When he missed a simple forehand during the second-set tie-break, Djokovic pounded his racket against the ground before stomping off to unwrap a new one.
The Serbian is seeded 20th following his recent absence through injury, which necessitated a minor operation to ease an elbow problem earlier this year.
Yet the elbow certainly did not seem to be troubling Djokovic as his racket bore the brunt of his irritation.
Nevertheless, the 2016 Paris winner eventually found a way past Bautista Agut, winning 6-4 6-7 (6/8) 7-6 (7/4) 6-2.
And his route deeper in the tournament may have been made a little easier, too, after fourth seed Grigor Dimitrov bowed out.
Djokovic will instead meet another Spaniard, Fernando Verdasco, in the last 16.
Bulgarian Dimitrov won the ATP Tour Finals in London last year, but his wait for a grand slam title will go on after a 7-6 (7/4) 6-2 6-4 loss to the 30th seed.
Verdasco has now reached the fourth round at Roland Garros seven times, but he has yet to progress any further.
Second seed Alexander Zverev, along with Djokovic seen as one of the few genuine challengers to 10-time champion Rafael Nadal this year, was taken to five sets for the second time.
In the match of the tournament so far, the 21-year-old German was up against it trailing 2-1 to Damir Dzumhur.
Amid some spectacular rallies Dzumhur was playing shots far above his ranking of 26, and the Bosnian forced two match points in the deciding set.
But he put the second, a simple forehand winner, into the net and Zverev survived before going on to register a 6-2 3-6 4-6 7-6 (7/3) 7-5 victory in just under four hours.
— Eurosport UK (@Eurosport_UK) June 1, 2018