The 11th seed Tiafoe appeared to have lost all the magic of his early campaign before wresting the momentum in the third set and clinching it with a forehand winner to the deafening cheers of the fans at Arthur Ashe Stadium.
Michelsen had flown under the radar in his deepest major run but earned respect on the biggest stage of American tennis with 21 aces, and Tiafoe cut his own celebration short to comfort the 22-year-old in an emotional embrace.
"I slowly, slowly started getting my rhythm there," said Tiafoe, who will next play either eighth seed Ben Shelton or defending champion Carlos Alcaraz. "He's a hell of a player, man."
Tiafoe thrived in his beloved Louis Armstrong Stadium this week but struggled to hit his stride at Ashe, sending four winners versus his opponent's 15 in the opening set, before handing over the critical break on set point.
Michelsen did some of his best work running up on the net in the second set, where he got the decisive break with a fine backhand strike, and he sent another shot down the line to break in the opening game of the third.
The unseeded Michelsen was serving for the match when his nerves took over and he put up two double faults. It was the opportunity that Tiafoe was looking for and the veteran dug in his heels, converting on break point to draw the set level at 5-5.
Tiafoe slammed his foot on the gas, surviving a cat-and-mouse exchange for another break on set point, before winning the first three games of the penultimate set as he upped his level across the board.
The two brought their best tennis to a nerve-searing final set, where the crowd sprang to their feet as Tiafoe sealed the win before offering words of encouragement to a weeping Michelsen.