Double grand slam winner Murray needed just one hour and 24 minutes to send world number 92 Rola packing, a 6-1 6-1 6-0 victim.
The 27-year-old Scot will tackle Spaniard Roberto Bautista Agut in the third round, after the world number 23 saw off Czech Jan Hernych 7-5 4-6 6-2 6-2.
Top seed and title favourite Novak Djokovic didn’t have things as easily and he had to dig deep into his reserves to fend off a determined Radek Stepanek and reach the third round .
A win against Murray and run to the semi-finals at the Aegon Championships at Queen’s Club marked Stepanek down as a player in form on grass, and he tested Djokovic close to the limit.
Eventually the Serbian found a way to beat perhaps the most dangerous of the unseeded players in the draw, coming through 6-4 6-3 6-7 (7/5) 7-6 (7/5) on Centre Court.
Djokovic said: “I was two sets up and had some break-point chances in the third and I should have closed it out in the third set tie-breaker, but credit to him for fighting.
“He’s 35 years old but he’s moving very well. Grass is probably his most preferred surface. He performs really well on the big stages as yousaw.
“He loves to engage the crowd, he’s an entertainer, and it was fun from one side to be part of a great thrilling match, but on the other side I should not have complicated my own life this way.
“It was annoying I came to Wimbledon without any official match practice and my first-round match went just over hour, so I didn’t have a lot of match play.
“I was quite focused and tense before the match because I knew what was on the other side of the net. I knew his ability and his quality.
“It was a difficult one but I’m glad I stayed hanging in there mentally and managed to win.”
David Ferrer’s run of 10 consecutive grand slam quarter-finals ended when he was beaten in the second round at Wimbledon by Andrey Kuznetsov.
The Russian, who won the junior title at the All England Club five years ago, edged a tight contest 6-7 (5/7) 6-0 3-6 6-3 6-2 for his first ever five-set win and first victory over a top-10 player.
It was a hugely impressive win from the 23-year-old, ranked 118th, not least because of Ferrer’s consistency at the slams in recent years.
The seventh seed lost before the last eight at a slam for the first time since the US Open in 2011, and it was his earliest defeat for four and a half years.
Latvia’s Ernest Gulbis was another big hope to crash out, the 12th seed losing out to world number 90 Sergiy Stakhovsky 6-4 6-3 7-6 (7/5).
Big-serving South African Kevin Anderson moved past Edouard Roger-Vasselin in four sets, a 7-6 (7/0) 1-6 6-3 6-4 winner.
Ukraine’s Alexandr Dolgopolov dismissed German Benjamin Becker 6-7 (7/4) 7-6 (7/0) 6-3 6-4, the 21st seed marching on with little issue.
In the women’s singles, second seed Li Na breezed past Yvonne Meusburger in just over an hour on Court Two.
The 2014 Australian Open champion disposed of Austrian Meusburger 6-2 6-2.
She now faces Czech Barbora Zahlavova Strycova after she knocked out the 32nd-seeded Russian Elena Vesnina 6-4 6-2.
Number four seed Agnieszka Radwanska of Poland also moved safely into the third round as she recorded a straight-sets win over Australian Casey Dellacqua in the first match on Centre Court.
Five-time Wimbledon champion Venus Williams survived a scare to fend off the challenge of Japan’s Kurumi Nara and come through 7-6 (7/4) 6-1 on Court Two.
The American will next face sixth seed Petra Kvitova after the Czech player, champion in 2011, won 6-2 6-0 against Germany’s Mona Barthel.