Haase saved all six break points against he faced and improved to 11-0 for the event.
"The match was less one-sided than the result suggests," the Dutchman said. "He had a lot of chances but I played good points when it mattered most. I don't feel pressure as I've already proved last year I am able to defend the title."
Brands, who beat Roger Federer in the second round in Gstaad last week, dropped his serve four times.
Haase will next play Verdasco, who won an all-Spanish match against Guillermo Garcia-Lopez 6-2, 7-6 (7).
Thiem reached his first career quarterfinal after recovering from a viral illness, which sidelined him for six weeks. He converted the only break point to take the opening set and led 5-0 in the second when Melzer won three straight games and missed three break points for 5-4.
"I didn't play well at all and Dominic deserved to win," Melzer said. "I served pretty well in the first set but my returns were awful all match long."
Monaco beat Austrian wild-card entry Andreas Haider-Maurer 6-3, 6-4. The Argentine, who won the event in 2007, came back three times from a break down in the second set.
"I have good memories from six years ago but this is a totally different event," Monaco said. "I've changed a lot and I am far more experienced now."