Djokovic, who finished runner-up to Alcaraz at the Australian Open in January, won 87% of his points on serve to clinch the opening set.
But Draper stepped up level to force a decider as quality improved and Djokovic won a tough 26-all rally in the opening match 30-30 on the Serb’s serve.
The Englishman broke his rival’s next service game to take a 5-4 lead but was broken when the match went to a tie-break.
Once again, Draper pulled off an impressive regroup to advance to the quarterfinals against former world number one Daniil Medvedev.
“It was a real physical fight and there were a lot of twists and turns in the fight,” said Draper, 24.
“Getting through that game is something I’ll never forget.”
Djokovic, playing in his first tournament since reaching the Australian Open final in January, believes he paid the price for the 26-stroke rally.
“I needed some time off after that,” he said. “It was great to win that point in the game, but I really ran out of steam.
“The crowd supported me and I felt the energy. I felt like, ‘Maybe I can take this.’”
“He’s playing at a great level. Overall, he’s a great player and a great person.”
