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"I've played some great tennis last couple of months," Rublev said. "We'll see if I can keep going."
The 23-year-old from Moscow will next face Croatia's 32-year-old Marin Cilic, who beat Italy's 19-year-old Lorenzo Musetti 6-3, 6-4 in 74 minutes.
"Very pleased with today," Cilic said. "Just have to battle it out with these youngsters. I've got to keep the heads high for the veterans on the tour."
World number 45 Cilic, the 2014 US Open champion, feels he is working his way back into match fitness.
"I felt really well in the practices. The off-season went really well," he said. "Just missing matches. It just hurts the game a bit."
Canada's Milos Raonic, ranked 19th, advanced with a 6-4, 7-5 victory over France's 30th-ranked Ugo Humbert.
He will next face Polish 26th seed Hubert Hurkacz, who dispatched Canadian sixth seed Denis Shapovalov (CAN x6) 6-3, 7-6 (8/6).