Johnson, a two-time NCAA singles champion from the University of Southern California, saved seven breaks in the first set and took advantage of a double-fault by Haas in the third set to come away with a 6-4, 2-6, 7-6 (7-2) win and a trip to the quarterfinals.
Johnson will face sixth-seeded Feliciano Lopez of Spain in the next round. Lopez rallied to beat Frenchman Adrian Mannarino 4-6, 6-4, 6-2.
Marinko Matosevic also advanced when eighth-seeded Lleyton Hewitt retired with an injured right shoulder. Fourth-seeded Kevin Anderson of South Africa moved on when Croatia's Ivo Karlovic retired with a stomach ailment. Seventh-seeded Marin Cilic of Croatia ousted Benjamin Becker of Germany 6-1, 6-3.