The fourth-seeded Czech, twice a winner in Gstaad, was outplayed by claycourt specialist Montanes in the opening set before he took control to complete his 12th straight victory in the event.
Novak, who beat Wimbledon champion Roger Federer in last year's final, won the title for the first time in 2001.
"I struggled in the first set but was more aggressive in the second," said Novak. "In the third, I was really focused."
Novak will play Potito Starace in the quarter-finals after the unseeded Italian beat Romanian Victor Hanescu 3-6, 7-6, 7-6.
Russian Igor Andreev defeated 1999 tournament winner and eighth seed Albert Costa of Spain 6-2, 6-4 to book his place in the last eight.
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The Russian will meet Ruben Ramirez Hidalgo, for a place in the semi-finals, after the Spaniard beat Austrian Stefan Koubek 6-3, 6-4.
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