Seventh-seeded Rune, who is ranked 9th in the world, had most trouble in the first set.
Michelsen broke the Dane and took a 3-1 lead, but Rune responded immediately and restored parity at 3-3.
With the score 4-4 and 40-0 in Michelsen's serve, Rune had three break opportunities, but the American managed to save them all.
The first set almost inevitably ended in a tiebreak, where Rune showed his ability to find his tennis at the decisive moment, while frustrations began to build up for Michelsen.
In the second set, Rune immediately broke Michelsen to take 2-0 lead and a little later extended his lead to 3-0 oozing confidence in all aspects of the game.
All energy seemed to seep out of the frustrated Michelsen who also seemed to struggle with the Cincinnati heat.
Leading 5-2, Rune missed a match point in the American's serve, but the Dane served the match home himself when he took advantage of his third match point.
Rune, who reached the semi-final of last year's tournament, will now face the winner of the match between American Frances Tiafoe and Frenchman Ugo Humbert. Tiafoe is number 14 in the world, while Humbert is ranked 25.