World number one Carlos Alcaraz sailed into the last eight of the Masters 1000 event for the fifth straight year and there were straight-sets wins for Daniil Medvedev and Cameron Norrie.
But Britain's Draper did it the hard way, wearing down 38-year-old Djokovic in a punishing third set to deny the Serb superstar his first return to the quarter-finals since he won his fifth Indian Wells title in 2016.
"I came out here tonight and I won that match through determination and trying to problem-solve and do my best and have a great attitude," said Draper, who kept the former world number one on the move with multiple drop shots.
The margins were razor-thin over the first two sets. The tide turned on an epic point in the opening game of the third that saw both players chasing down drop shots and scrambling for lobs before Djokovic sealed it with an overhead for a 40-30 lead.
He flopped on the court exhausted and was on his knees again after Draper won the next point. Djokovic would go on to hold serve, but he said it was the difference in the match for him.
"One point," he said. "It was great winning that point in that game, but I just ran completely out of gas."
Draper broke Djokovic in his next service game, but couldn't close it out when he served for the match at 5-4.
The reprieve wasn't enough for Djokovic, who led 4-3 in the tiebreaker but couldn't hang on.
"He played a sloppy game to close it out 5-4, and, you know, I got the crowd, you know, backing me, and I felt the energy," Djokovic said. "It was, like, maybe I'm gonna take this one. It was so, so close."
Draper, playing just his second tournament since missing the better part of six months with an arm injury, was delighted.
"To come out here against Novak, for me the greatest tennis player there is, I'm just incredibly proud of myself," he said.
He won't have long to savor the victory, with a quarter-final against two-time finalist Daniil Medvedev coming up on Thursday.
Medvedev beat American Alex Michelsen 6-2, 6-4.