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.
Djokovic said: “Right now, considering I played almost four hours in my last match, I felt pretty good physically. I had no problems going the distance in long rallies and I won in three sets against a player who is in form, especially for this surface.”
Djokovic will face Marco Cecchinato after the Italian pulled off a shock win over David Goffin.
Eighth seed Goffin, perhaps weary after two five-setters already this week, dropped the first set but hauled himself level at 1-1. However, Cecchinato, the world No 72, went on to win 7-5, 4-6, 6-0, 6-3.
Alexander Zverev has become marathon man of Roland Garros after coming through another five-setter to book his place in the last eight.
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 quarter-finals of a grand slam.
He has now played 18 sets to Nadal’s nine, with the Spaniard facing unseeded German Max Marterer today 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.”
Zverev’s opponent in the last eight will be Austrian seventh seed Dominic Thiem, who overcame Kei Nishikori of Japan.
Nishikori, the 19th seed, had hauled himself back into the match by snatching the third set 7-5, but threw his racket to the floor after he was broken for 4-3 in the fourth.
Thiem, 24, went on to register a 6-2, 6-0, 5-7, 6-4 victory.