Harrison, ranked 62nd, will jump into the world top 50 for the first time since July 2012 by taking the matchup of unseeded 24-year-olds who had not dropped a set all week at the indoor hardcourt event.
The breakthrough victory came after Harrison had lost seven ATP semifinals but downed compatriot Donald Young to reach the Memphis championship match.
Harrison blasted eight aces in downing Basilashvili, who will rise from a career-best 67th in the rankings despite a loss in his second career final.
Basilashvili, who made a semifinal run last week in Sofia, lost his only prior ATP final to Italy's Paolo Lorenzi at last July's Austrian Open.
Harrison saved two break points before holding in the opening game of the match, then broke the Georgian in the fourth and sixth games for a 5-1 lead and held at love to claim the first set in 26 minutes.
