The 35-year-old Haas, who won the event in 2009, is coming off a quarterfinal appearance at the French Open.
After losing the first set to a forehand winner by Haas, Gulbis broke his racket in anger and decided to change his shoes and shirt as well.
In the second set, Haas broke to lead 3-2 and stayed ahead for good. The German never faced a break point and overcame 15 aces by his Latvian opponent.
"Overall, I played a good match. But I would have been very upset if at the end I'd lost because I couldn't take all those chances," said Haas, who had many break-point opportunities in the match.
The oldest player in the top 100, Hass is aiming for his second title of the year. The 11th-ranked German won last year's title by beating Roger Federer.
"He returns very well, he doesn't give you much space to maneuver him around the court," Gasquet said of Mayer.
The Frenchman was pleased with his match after a bye into the second round.
"I served big and I returned well," said Gasquet, who landed 88 percent of first serves. "The grass here is faster than in England."