For former college champion Rola, the 6-1 6-1 6-0 defeat was a painful lesson in what it takes to be a grand slam winner.
Murray, with victories at the 2012 U.S. Open and last year's Wimbledon under his belt, has his sights set on the bigger prize of a successful title defence this week and has no time for sentiment.
"You try to finish matches as quickly as you can. If you have the momentum with you and you're playing well, that's what you need to try to do," the British third seed said.
At this month's French Open, where Murray lost to champion Rafael Nadal in the semi-finals, the Scot said he was sometimes not clinical enough.
"Some of the matches where I was ahead, I didn't finish the sets off as best as I would have liked. So I wanted to make sure here that when I had the momentum, when I was on top, that I finished the sets off. I did that well today," he said.
While the British No.1 was winning Wimbledon last year, Rola was winding up a successful college tennis career for Ohio State University. The gulf in quality was stark on Wednesday.
But Murray said he had to treat every player as a potential danger.
"When you are in a position to win a match like that, you have to try to do it as quickly as possible, because all the players in this tournament are very, very good tennis players," he said.
"If you give them a look-in in a set or they see a way back in, they can start playing very well."
Murray gave Rola no way in and on an otherwise partisan Court One some of the biggest cheers were reserved for the 23-year-old Slovenian's two winning games, the crowd willing him to avoid further embarrassment.
But Murray, blasting winners from both sides of the court, put the 23-year-old firmly in his place to set up a third round clash with 27th seed Roberto Bautista Agut of Spain.
Murray practised with Bautista Agut earlier in the season and knows him well.
"He's a very good player. He won the tournament last week on the grass in Holland," Murray said.
"He doesn't play like a lot of the Spanish guys. He plays very flat. Not much topspin. The grass courts suit his game pretty well. It will be a step up for sure."
(Editing by David Goodman)
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