Draper won 6-0 6-4 in one hour and twenty-six minutes in Acapulco by holding serve the entire time and making four of his five break-point chances.
Next up for him is lucky loser Yoshihito Nishioka of Japan.
Dan Evans, Draper’s countryman, was eliminated after falling to American Ben Shelton 2-6 7-5 7-6 (7-5) in the finals.
In the third set, Evans survived three match points and served for the match, but Shelton, the world number seventeen, hit back.
It will be “definitely a really tricky match-up for me,” Shelton remarked. “I’m really familiar with Dan; we joke around a lot in the locker room, and he has a really difficult game.
“So to be able to come out of this first match here with a win after losing the first set is a big accomplishment for me.”
