Djokovic broke for 3-1, seeing out the decisive moments in a set where he won only three more points than his opponent.
Verdasco made 48 unforced errors and grew increasingly furious with himself as he missed out on a first French Open quarter-final at the 16th attempt.
But Zverev, as he did in his wins over Dusan Lajovic and Damir Dzumhur, forced a decider, sealing the set with an ace after saving two break points. The 36-year-old 23-times Grand Slam victor, returning to the game after maternity leave this year, was unable to play due to a pectoral muscle injury.
"For me is fantastic".
Eurosport analyst Patrick Mouratoglou looks at the strengths of Marco Cecchinato before he faces Novak Djokovic in the quarter-finals.
Djokovic won on his second match point with an easy smash at the net.
"I was a bit unlucky not to capitalize on the second set", said Djokovic, who improved to 7-1 against Bautista Agut.
Cecchinato is bidding to become the first Italian man to reach the semi-finals at a Grand Slam since Corrado Barazzutti in Paris in 1978.
He survived Tuesday by returning a Djokovic serve with a backhand just inside the baseline and then collapsing onto the court.
Cecchinato received a penalty point immediately after Djokovic was treated, for unsportsmanlike conduct after a lengthy argument with the umpire and supervisor for leaving the court without permission.
Djokovic sprinted out to 5-2 but was reeled back in to 5-5, having squandered three points for a double break in the sixth game.
Djokovic wrapped up the second set with a rasping forehand victor after his opponent was treated for a blister on his right foot and another errant backhand putVerdasco a break down in the first game of the third.
Alexander Zverev, the second seeded German, is in his first quarter-final at the majors where he faces Dominic Thiem who has made the semi-finals in Paris in the last two years. He arrived at Roland Garros with a 10-7 record this season.
"He is one of the fittest guys on the tour so it was hard for him today", said Thiem, the only man to have defeated 10-time French Open champion Rafael Nadal on clay this year.
There was no stirring comeback this time for Zverev on Court Philippe Chatrier. "I knew I'm not going to win the match". "But I still wanted to finish the match and give the credit to Dominic". There was no way for me. I couldn't serve. I couldn't really do anything. "He deserves to be in the semi-finals".