Murray, the world number two and champion in 2013, crushed Taiwan’s Lu Yen-Hsun 6-3, 6-2, 6-1, hitting 31 winners and serving six aces.
He will face Australian world number 67 John Millman for a place in the last 16.
“There were a lot of close games in the first set, but once I managed to hang on there I settled down,” Murray said.
Muguruza, the French Open champion and last year’s runner-up, lost 6-3, 6-2 to Slovak qualifier Jana Cepelova, the world number 124, in just 59 minutes on Court One.
It was the same arena where the 23-year-old Cepelova had defeated Simona Halep last year, a result which illustrated the Slovak’s liking for the big time — having also seen off Williams in Charleston in 2014.
“I deserved to win. I’m so happy,” said Cepelova.
Spanish 22-year-old Muguruza was bidding to become only the eighth woman to win the French Open and Wimbledon back to back.
But she was undone by 22 unforced errors and hitting just nine winners in a flat performance.
“I think my energy was missing a little bit today. From yesterday I felt already a little bit tired, I think is the best word,” said Muguruza.
In all, 11 seeded women exited the tournament as well as six men’s seeds, the highest of which was number eight Dominic Thiem of Austria.
Thiem lost in three tie-break sets to Jiri Vesely of the Czech Republic.
Serbia’s Viktor Troicki made his case for the greatest Wimbledon rant of modern times as he slipped to a five-set defeat to Spain’s Albert Ramos Vinolas.
Infuriated that umpire Damiano Torella had over-ruled a Vinolas serve which was called out to declare it an ace, Troicki unleashed a tirade of abuse at the Italian official.
“You’re the worst umpire in the world, you’re so bad,” screamed Troicki as he went match point down and he continued in the same vein when the match concluded.
“You’re horrible, you know what you did. You are an idiot.”
France’s Gilles Simon, who lost to Grigor Dimitrov on Court One, was also letting off steam, threatening to sue the organisers for making him to play in the rain.
“I won’t play when it rains. I know you have commitments to TV but if I get injured I will sue you and I will win,” Simon told the umpire before the brief shower passed on.
Poland’s third seed Agnieszka Radwanska, the runner-up to Williams in 2012, saved three match points to beat Croatian teenager Ana Konjuh, 6-2, 4-6, 9-7.
The 18-year-old’s challenge ended in tears when she needed a medical time-out at 7-7 in the final set after turning her ankle badly when she stepped on the ball.
Radwanska next faces Katerina Siniakova for a place in the last 16.
Nineteen-year-old Belinda Bencic, the Swiss seventh seed, retired from her second round match against US qualifier Julia Boserup.
Bencic was 6-4, 1-0 down on Court Three to the world number 225 when she quit with a wrist injury.
Five-time women’s champion Venus Williams ignored her exile to Court 18 to battle past Greek qualifier Maria Sakkari 7-5, 4-6, 6-3.
“It’s not the ideal schedule for the women,” said 36-year-old Williams when asked about being shifted to Court 18.
“We’d like to see equal amount of matches. We don’t want more, just the same amount.”
German fourth seed and Australian Open champion Angelique Kerber beat Varvara Lepchenko of the United States 6-1, 6-4.
There were also second round wins for fifth seed Halep of Romania and US ninth-seed Madison Keys.
Japan’s Kei Nishikori saw off France’s Julien Benneteau, whose world ranking has sunk to 547 due to injury, 4-6, 6-4, 6-4, 6-2 and goes on to meet Russia’s Andrey Kuznetsov.
Canadian sixth seed Milos Raonic brushed past Andreas Seppi 7-6 (7/5), 6-4, 6-2.