The home favourite, who last year ended Britain's 77-year hunt for a men's singles champion, flattened the Slovenian 6-1 6-1 6-0 to reach the third round without dropping a set and stretch his winning streak at the All England Club to 15.
Veteran American Venus Williams, another player who thrives at the spiritual home of lawn tennis, if not in recent years because of illness and injury, battled into the third round with 7-6(4) 6-1 defeat of Japan's Kurumi Nara.
Williams, level with younger sister Serena on five Wimbledon singles titles, missed last year's tournament and exited in the first round in 2012 but the 30th seed hopes she can at least join her sibling in the second week this time.
"I guess I haven't held up my end of the bargain," the 34-year-old told reporters of her struggles. "I tried. I just haven't had the luck I've wanted."
Women's seeds flourished in cool, sunny conditions early on Day Three with world No.2 Li Na untroubled by Austria's Yvonne Meusburger and 2012 runner-up Agnieszka Radwanska, seeded four, opening action on Centre Court with an easy win against Australia's Casey Dellacqua.
Twelfth seed Ernests Gulbis's hopes of making an impact following his semi-final run at Roland Garros ended in defeat to Ukraine's Sergiy Stakhovsky 6-4 6-3 7-6(5).
The Latvian is the highest-ranked men's casualty so far although defeat by the net-rushing Stakhovsky, the man who knocked out seven-times champion Roger Federer at the same stage last year, was not a complete surprise.
"Two of my biggest weapons, my serve and return, wasn't there," said the 25-year-old.
Murray is unbeaten at Wimbledon since the 2012 final against Federer. A few weeks after that emotional defeat he returned to win an Olympic gold medal and last year he scaled the heights to win the grasscourt grand slam against Novak Djokovic.
An early defeat at the warm-up tournament at nearby Queen's Club raised a few doubts over Murray's form coming into Wimbledon but those were cast aside in a one hour 24 minute masterclass against the unfortunate Rola.
Murray treated fans on Court One to some delightful shot-making, whether a backhand pummelled on the line, a delicate lob or the grass-hugging slices that drive far better opponents than Rola to distraction.
The 23-year-old Rola, playing at his maiden Wimbledon and against a top-20 player for the first time, had the temerity to earn a break point in each of the first two sets, both times Murray responding with a booming ace.
Murray, who endured some roller-coaster rides during his run to the French Open semi-finals, was clearly in no mood to work any overtime as he accelerated to victory.
"Getting games finished quickly helps," Murray said after coming off court.
"I feel like that cost me a little bit at the French Open - not that I would necessarily have beaten Rafa (Nadal), but I played a lot of sets and in a lot of those sets, I was up."
Petra Kvitova, another former champion, was also in a hurry, beating Germany's Mona Barthel 6-2 6-0. The Czech faces a heavyweight third round against Venus Williams.
"All of our matches have always been tough," Williams said. "I know I'll have to get out there and play better than her. That's really what it boils down to."