The British No4 silenced the home crowd with a gritty display in a 6-4 6-2 3-6 6-3 victory.
The Grandstand court at Flushing Meadows is an intimidating place for an overseas player taking on an American, but Draper had almost emptied it when he went 2-0 ahead.
Mmoh, ranked 89 in the world, hit back in the third, but Draper dug deep to break for 4-2 in the fourth before clinching a huge win.
Grandstand is also the court which Draper hurt his hamstring a year ago in his third-round match against Karen Khachanov, forcing him to retire.
The 21-year-old has been beset by injuries ever since and was a doubt to even play in New York due to a tear in his shoulder.
Yet Draper, serving with less vim than usual in a bid to manage the problem, still thumped 52 winners to surge into the last 16.
He said: "It was a tough match, Michael is an incredible player who has had a great year and I knew would be a challenge. I was proud of the way I came through it.
"Last year was really difficult, I injured myself and didn't want that to happen today. To come it through after a tough year for me, I'm very happy."
British No1 Cameron Norrie tumbled out in the third round after a straight-sets defeat to Matteo Arnaldi. An out-of-sorts Norrie, the 16th seed, was beaten 6-3 6-4 6-3 by the Italian world No61.
“I actually think I started well,” he said. “I was hitting the ball really well all week but I didn't have the shot tolerance to hang with him. He was really crafty and won a lot of tough points so credit to him.”
Katie Boutler is also out after a straight-sets defeat to Peyton Stearns while Carlos Alcaraz knocked out Dan Evans.