Johnson, who is 23 and played college tennis at Southern California, beat Tennys Sandgren 4-6, 6-3, 0-6, 7-6 (5), 6-1 in the men's final on an indoor hard court.
Johnson is ranked 156th. His best showing at a major tournament was reaching the third round at the 2012 US Open as a wild-card entry.
The 195th-ranked Vickery defeated Victoria Duval 6-2, 6-3 in the USTA playoff women's final. Vickery also earned a wild card into this year's US Open, her Grand Slam debut, and got to the second round.
The Australian Open starts Jan. 13.