Draper wore down the 38-year-old Serb superstar in a punishing third set, denying Djokovic his first return to the quarter-finals since the last of his five Indian Wells titles in 2016.
Djkovic, a 24-time Grand Slam champion playing his first tournament since falling to Carlos Alcaraz in the Australian Open final, looked to be feeling the strain as the final set unfolded.
Draper, playing just his second event since missing the better part of six months with an arm injury, kept Djokovic moving with a stream of drop shots.
Draper was broken as he served for the match in the ninth game, but Djokovic's second wind wasn't enough and Draper will take on 11th-seeded Russian Daniil Medvedev for a place in the semi-finals.