The 22-year-old, who beat seventh seed Tommy Paul in the previous round, proved too strong for Japan’s Yoshihito Nishioka, winning 6-3 6-0 in 68 minutes.
The British number three started strongly, breaking the world number 82 in his opening scoring game and threatening as he opened a 3-0 lead. Games went on serve as Nishioka fought to stay in touch at 5-3, but Draper served out to win the opening set in 42 minutes.
He was barely troubled after that, again breaking Nishioka to start the second set. He broke twice more and stayed firm on his own serve to book a place in the last eight.
Draper will face Miomir Kecmanovic, ranked three places below him at 53, after the Serbian beat Daniel Altmaier 6-4 6-4.
Boulter needed more than two hours and 20 minutes as she came from a set down to beat second seed Beatriz Haddad Maia 3-6 6-3 6-4 in the San Diego Open. The British number one had three break points midway through the first set, only for the Brazilian world number 13 to fight back and take her only break point of the first set.
Roles were reversed in the second set as Boulter claimed the only break and raced into a 5-1 lead in the deciding set.
Haddad Maia broke back and cut the deficit to a single game, but Boulter made no mistake serving for the match a second time to seal her first quarter-final at WTA 500 level.
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