Bedene served for the first set at 5-4 but Wawrinka won the next three games to take the set. The Swiss player then saved four break points in the second.
"I'm happy with the first win. I think it was a great level," said Wawrinka, who will next face fourth-seeded Karen Khachanov.
Wawrinka, a three-time Grand Slam champion, missed the end of the 2017 season and much of 2018 with a knee injury. He is now ranked 88th in the world and made the third round at the U.S. Open.
Also, Martin Klizan beat Evgeny Donskoy 6-4, 6-4.
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