Murray, the third seed, coasted to a 6-1, 6-1, 6-0 mauling of 23-year-old Rola who was playing just the 10th match of his career having only graduated from university in Ohio last year.
The British star eased to victory in just 84 minutes, hitting seven aces and 27 winners against the world number 92 and goes on to face either Jan Hernych of the Czech Republic or Spanish 27th seed Roberto Bautista Agut for a place in the last 16.
"It's his first year on grass so it was tough for him. But he has a good game and he'll keep improving," said Murray.
"I took care of my serve for the majority of the match and I played well. If you can finish matches as quick as possible then that's better in the long run."
The colourful Gulbis then denied he had lost all of his $500,000 winnings from the French Open in a Riga casino.
"I went to play blackjack, but there was no word how much I won or how much I lost. They (reporters) asked me how much I lost. I said, A lot. I was joking," he said.
"Next day in Latvia they say I lost all my winnings. My mother sends me a message next day asking am I crazy. I may be but I'm not stupid."