Edmund, who was cheered on by a vocal home support, shocked Djokovic to claim the opening set 6-4 after 49 minutes of tennis in the baking hot English capital.
The 23-year-old continued to impress against his more experienced opponent in the second set, but Djokovic secured a break at 3-4 before serving it out to level the contest.
Djokovic started to come into his own in the third set - breaking Edmund's first serve before securing another break at 2-4 before serving it out to move 2-1 ahead.
The British number one recovered to hold his first four service games of the fourth and what proved to be final set, but Djokovic secured a vital break at 4-4 before serving out the contest to move into the fourth round of the competition.
Three-time champion Djokovic, who is seeded 12th for this year's competition, will play Russia's Karen Khachanov in the last-16 stage of the competition on Monday.