Djokovic, who won his only Roland Garros title in 2016 to complete a career Grand Slam, was not considered among the pre-tournament favourites following an injury-hit year.
"Considering all the circumstances in the last 15 months it is quite an achievement to get to the quarters of a Slam," said Djokovic, who has also reached the Roland Garros last eight for a ninth successive year.
"I always play well at a Slam and always build my form to peak at a Slam.
"Hopefully it will get even better. I don't want to stop here."
Djokovic easing through the draw
Former world number one Djokovic has dropped down the rankings to 22nd after taking time off with an elbow injury following last year's Wimbledon. He also needed a minor operation after making a comeback at the Australian Open in January.
However, he is quietly going about his business in Paris and has dropped only one set on his way to the last eight.
Djokovic may still not playing at the same level as he was at the peak of his powers, but his movement and defensive ability - two of his great hallmarks - look good enough to trouble most opponents.
Greater tests than Verdasco, who had beaten Djokovic in three of their previous four meetings on clay but not since 2010, will lie ahead.
If Djokovic beats Cecchinato he will meet either German second seed Alexander Zverev or Austrian seventh seed Dominic Thiem - who have won more Tour matches in 2018 than any other player - in the last four.
He would not face 10-time champion Rafael Nadal - a rematch of the 2012 and 2014 finals - until the final.