Eleventh seed Tiafoe overcame a 0-2 deficit and advanced to the quarter-finals after a tiebreak in a match that lasted 281 minutes.
The match was 22-year-old Michelsen's first grand slam quarter-final, and when it was over he cried out his frustration on Tiafoe's shoulder.
In the semi-finals, Tiafoe will face either reigning champion Carlos Alcaraz or Ben Shelton.
