Djokovic, runner-up to Alcaraz at January’s Australian Open, won 87% of points on his serve as he clinched the opening set.
However, Draper levelled to force a decider as the quality rose, with Djokovic winning a gruelling 26-shot rally in the opening game at 30-30 on the Serb’s serve.
The Briton broke his rival’s next service game but, serving for the match at 5-4 up, was broken back as the match went to a tie-break.
Once again, Draper regrouped impressively to advance and set up a quarter-final against former world number one Daniil Medvedev.
“It was a real physical battle, there were definitely a lot of ups and downs in the match,” Draper, 24, said.
“To come through that match is something I will never forget.”
Djokovic, playing his first tournament since reaching the final of the Australian Open in January, believes he paid the price for the 26-shot rally.
“It cost me a break after,” he said. “It was great winning that point in that game, but I just ran completely out of the gas.
“I got the crowd backing me and I felt the energy. It was, like, ‘maybe I’m going [to] take this one’.
“He is playing at a great level. Just, overall, a great player, great guy.”
