Dimitrov, the 2017 ATP Finals champion, conceded his serve once but broke his rival's serve, forcing the set into a tie break.
Three set points came and went in the 12th game of the second set, three break points were wasted in the sixth game of the fourth before he blew the chance to level the tie at two sets apiece in the ninth game.
He would not face 10-time champion Rafael Nadal - a rematch of the 2012 and 2014 finals - until the final.
Still, Djokovic did appear to be gathering momentum, taking the third set and moving out to a lead in the fourth.
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Seeded 20th after a well-documented slump, Djokovic blunted Verdasco's explosive firepower with his signature counter-punching to book an unlikely last-eight date with Marco Cecchinato.
"I just could never turn it around out here".
Earlier in the day, second seed Alexander Zverev finally weared out after three days of clay battle and Dominic Thiem defeated him to make it to the Roland Garros semi-final for the third consecutive year.
Zidane steps down as Real Madrid Coach
As part of United, Ronaldo had won three titles and the Champions League before he joined the Spanish team in 2009. Zidane was hired in January 2016, his first major coaching job.
In the first round at Roland Garros, he dropped the first two sets against Marius Copil, a player ranked 94th, before coming all the way back to win 10-8 in the fifth.
"It is hard. Many things in life are hard", said Djokovic.
"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". "So it's going to be an fantastic challenge for me".
Djokovic won 144 points to Cecchinato's 140 in the end but was surprisingly less effective in the clutch plays, one particular area of his game that seemingly still needs recovery.
Djokovic served for the fourth set at 5-3 but got broken.
The 12-time grand slam victor needed lengthy treatment on a neck problem after dropping the first set.
German second seed Alexander Zverev and Novak Djokovic edged closer to a possible French Open semi-final clash of generations, with the youngster edging out Karen Khachanov and the former champion seeing off Fernando Verdasco in the last 16 on Sunday.
Nothing should be taken away from Thiem's victory, however, after he carved an easier route to the quarters and is rewarded for his work with a spot in the final four, where he'll face Cecchinato.
Kazakhstan's Putintseva will be playing her second quarter-final at Roland Garros, but this time it has been more of a surprise after slipping to 98th in the world rankings.