One break of serve was enough to take the opening set and although left-hander Verdasco produced thunderous winners to stay in touch in the second his challenge soon faded.
Djokovic lost his temper after missing a simple forehand that would have given him set point, and he came close to losing the third set when 5-3 down.
Djokovic won the next two sets in a manner akin to the dominance he enjoyed prior to an elbow injury and loss of motivation, which took a huge toll on his form.
Roberto Bautista Agut has backed "great champion" Novak Djokovic to be "on the top again" after their epic third-round tussle at the French Open. "Every possible level of the colour of the rainbow, you go through it", he said.
Spain's Fernando Verdasco awaits in the fourth round after he stunned Bulgarian fourth seed and Association of Tennis Professionals World Tour Finals champion Grigor Dmitrov with a 7-6, 6-2, 6-4 win.
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 the Russian sealed the set at the third time of asking, leaving the World No. 3 needing a fightback for the third consecutive match.
"It was a big point and in these kind of circumstances emotions get the better of you", he said.
It could well have been an all-Spanish fourth-round match if Bautista Agut had succeeded in serving out the third set, which Djokovic pinpointed as the turning point. Djokovic advanced to his 40th Grand Slam quarterfinal. "I'm not proud of doing that, to be honest".
"He served for the match and I had to save lots of break points in the fourth and fifth sets".
Verdasco has now reached the fourth round at Roland Garros seven times, but he has yet to progress any further.
There were six breaks of serve in an erratic set before Djokovic clinched the tie-break shortly after the match clock passed the three-hour mark.
The world number 13 put the hard circumstances to one side as he focused on beating Djokovic for only the second time in their eight meetings.