But on number four Djokovic left a looping backhand return which landed just in and Cecchinato celebrated a famous victory, 6-3 7-6 (7-4) 1-6 7-6 (13-11).
Verdasco needed two hours and 21 minutes to advance past Dimitrov into the fourth round for the seventh time in his 15th participation in the tournament, reports Efe.
The Serbian reached for a drop shot to get a set point, this time.
Djokovic had also squandered three set points in the second set and failed to serve out the fourth in the ninth game after being 5-2 in front.
"I was two breaks down on the tie-breaker and I had two or three match points and then on the fourth one I was very exhausted". Analyzing the match, Djokovic added: 'He played wonderful and credit to him.
Spaniard Verdasco had earlier demonstrated his class on clay with a straightforward win over disillusioned fourth seed Grigor Dimitrov.
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Djokovic, seeded 20th, won 6-3 6-4 6-2 against 30th seed Verdasco.
Keys, the US Open runner-up to Stephens previous year, will face Kazakhstan's Yulia Putintseva for a place in the semi-finals, after the world number 98 saw off Barbora Strycova 6-4, 6-3.
"I'm just not thinking about tennis at the moment".
An elbow injury has limited the 12-time Grand Slam champion's playing time this year, while he was treated on his neck during the defeat by Cecchinato.
"It is hard. Many things in life are hard", said Djokovic.
The Italian, who has one career title to Djokovic's 68, had never won a match at the majors before this year's French Open.
Alexander Zverev of Germany celebrates winning against Karen Khachanov of Russian Federation during their men's round of 16 match during the French Open tennis tournament at Roland Garros in Paris, France. "I made that break at 3-1 and held my serve, and that helped because I knew that if we went the distance then I had a good chance", Djokovic said.
The storm seemed to have blown itself out and Djokovic served for the fourth set at 5-3.
To achieve that feat, Cecchinato, who in 2016 was suspended for 18 months for match fixing before his ban was lifted following a successful appeal, displayed an impressive range of claycourt tennis skills.
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.
World number 38 Khachanov has now lost at the last-16 stage at Roland Garros twice in a row, while Zverev ended his wait for a maiden major last-eight appearance at the 12th attempt.