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
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