Dolgopolov, the 22nd seed, is an unorthodox player who relies on spins and slices a lot more than power, and he had begun to entangle Djokovic in his web.
The world number one had to retrieve a break during the first set before finally coming through a 30-point tie-break on his sixth set point.
Djokovic celebrated as if he had won the match, and in many ways he had because Dolgopolov's threat diminished and the Serb was able to clinch a 7-6 (16-14) 6-4 6-2 victory - his 61st from 63 matches this season.
The clash was surprisingly played on Louis Armstrong Stadium, with queues of fans snaking around the block to get a glimpse of the world number one, and the windy conditions made life very difficult for both men.
Djokovic said: "It was an interesting first set. I think it was exciting for the crowd to watch because it was very close. But I think it was an ugly first set because I wasn't happy with the way I played.
"He was changing the pace a lot. He was always sending me a different ball. It was really hard for me to adjust to it because of the conditions that we played in.
"So that's why it was crucial for me to win the first set. After that I felt a bit more relaxed and I served well when I needed to, played okay, played well enough to win."
Djokovic had not played Dolgopolov before but unfamiliarity will certainly not be a problem with his next opponent, countryman Janko Tipsarevic, who reached the quarter-finals of a grand slam for the first time courtesy of a 7-5 6-7 (3/7) 7-5 6-2 victory over Juan Carlos Ferrero.
Tipsarevic and Djokovic are good friends and regular hitting partners, giving the eloquent Serb a unique perspective on the world number one's incredible season.
Tipsarevic, who is at a career-high ranking of 20th and has pushed Djokovic in both their previous meetings, said: "It's really a joke. I cannot believe that somebody lost two matches this year.
"You see guys on tour and, from a psychological point of view, they win a couple of matches in a row and you could feel in their body language on court that they feel a little bit satisfied.
"You really need to be a big, big champion in order to really do what you say in press conferences. Because you guys have probably heard it a million times: 'I'm going to focus and give 100% in my next match.'
"But that's not the case every time. The bottom line is that's great, but the goal is not to be overwhelmed by that. I'm playing against a good opponent. I know that I need to play aggressive and good in order to win.
"If I feel unbelievable and excited and just feel overwhelmed because I'm in the quarters for the first time and I'm playing Novak and I'm going out there to do my best, that's not going to happen.
"If you have an idea that you are going to win, the opportunities on the court which are going to be given to you, you're going to use them."
Djokovic said of the match: "It's a strange feeling. We are professionals. Certainly we both want to win the match when we play against each other. So you kind of forget about friendship. You put that aside.
"We've never played at a grand slam, we've never played best of five against each other, so it's going to be a first-time experience for both of us. He's playing the best tennis of his life. He's very confident."