The Briton, 21, overcame the big-serving American and 20th seed 6-4 3-6 6-2 7-6 (7-5) in two hours and 43 minutes on Court Louis Armstrong.
The world number 84 faces 2015 champion and top seed Djokovic on Sunday.
Djokovic reached the last 16 when a second successive opponent pulled out through injury at Flushing Meadows.
Edmund joined Johanna Konta in reaching the fourth round while compatriots Andy Murray and Dan Evans will on Saturday seek to join them in the last 16.
Edmund was beaten by Isner in straight sets at the French Open this year but the Briton's confidence has risen since he led Britain to Davis Cup victory, without Murray, over Serbia in July.
Isner, 31, fired down 27 aces but could only convert one of his seven break points as Edmund's serve held up under the pressure.
"It's been a great tournament so far," said Edmund, who knocked out world number 15 Richard Gasquet in the first round.
"I've really enjoyed playing here. It may not have been the result you guys wanted but I want to say thank for the support."
Perhaps the key moment came when the American had just taken the second set and created three break points at the start of the third.
Edmund, however, saved all three and then broke twice himself, sealing the set when Isner failed to make a simple forehand volley.
Isner saved two match points in the fourth set tie-break, but a forehand winner confirmed Edmund's thrilling victory.
"Chances don't come very often against John. You have to capitalise when they do. You see how many aces he hits," he added.
"You just have to get a racquet on the ball and hustle really.
"I played the big points well. I knew it would come down to one or two points. I'm just very pleased."
Collected