Murray, the second seed, breezed past Spanish veteran Tommy Robredo, the 32nd seed, 6-2, 6-4, 7-5.
The US Open champion goes on to tackle either Russian 20th seed Mikhail Youzhny, now the highest-ranked player left in his half of the draw, or Serbia’s Viktor Troicki for a place in the quarter-finals.
Murray, playing beneath the comfort of the Centre Court roof, fired 40 winners and nine aces past the 31-year-old Robredo.
“Upsets are never far away and you have to be very focused on your game to avoid that,” said Murray, who has seen Federer, Rafael Nadal, Jo-Wilfried Tsonga and Marin Cilic all removed from his path to the final.
“People are putting even more pressure on me because of the nature of how the draw has worked out.
“It would be a lot easier if we just concentrated on each match at a time.”
Stakhovsky, the world number 116, who sent seven-time champion Federer crashing out in the second round, went down 6-2, 2-6, 7-5, 6-3 to experienced Austrian Jurgen Melzer.
Instead of the Centre Court, where he had ended Federer’s run of 36 successive quarter-final appearances at the majors, Stakhovsky was exiled to Court Three where his challenge wasn’t helped by falling heavily on the slippery surface.
That required his ankle to be strapped and his challenge petered out as 32-year-old Melzer reached the fourth round for the second time where he will meet dangerous Pole, Jerzy Janowicz.
“I was possibly not prepared mentally because everybody expects you to play better after you beat Roger,” said Stakhovsky.
The 24th-seeded Janowicz made the fourth round for the first time by beating Spanish 15th seed Nicolas Almagro, 7-6 (8/6), 6-3, 6-4.
Spanish fourth seed David Ferrer reached the third round with a 6-3, 3-6, 7-6 (7/4), 7-5 win over compatriot Roberto Bautista Agut in match held over from Thursday because of rain.
“I had a little pain in my ankle but it’s not a problem,” said Ferrer, who made the last-eight in 2012.
Heavy rain delayed play on the outside courts with Bulgarian 29th seed Grigor Dimitrov staging a temporary sit-down protest to register his anger at conditions on Court Three before he lost a five-set, second round thriller.
Dimitrov, watched by superstar girlfriend Maria Sharapova, who was knocked out on Wednesday, went out 3-6, 7-6 (7/4), 3-6, 6-4, 11-9 to Slovenia’s Grega Zemlja.
When the tie resumed following Thursday’s suspension, 22-year-old Dimitrov slipped and fell in his service action at 8-9 in the final set to give up the third match point of the tie.
He then stomped off to the sidelines and sat in his courtside chair for 10 minutes waiting for the conditions to improve.
But it didn’t change Dimitrov’s luck as Zemlja took victory on a sixth match point.
“When I slipped, I fell down, I hurt my hip. I told the umpire, I’m not serving,” said Dimitrov.
Zemlja, the first Slovenian to make the third round at Wimbledon, goes on to face Argentina’s eighth seed Juan Martin del Potro for a place in the last 16.
Britain’s Laura Robson reached the third round of the women’s singles with a 6-4, 6-1 win over Colombian qualifier Mariana Duque-Marino and next meets Marina Erakovic of New Zealand.
“I’ve lost to her on grass before. She’s got a big game, a huge serve, a good slice as well. Pretty much a perfect game for grass,” Robson said.
Kaia Kanepi of Estonia, a quarter-finalist in 2010, knocked out German seventh seed, Angelique Kerber, who made the semi-finals last year, 3-6, 7-6 (8/6), 6-3.
“I had my chances in the second set but I didn’t take them,” said Kerber, who was 5/1 up in the tiebreaker.
The bottom half of the women’s draw has also been thrown open after world number two Victoria Azerenka withdrew with a knee injury and world number three, Sharapova slumped to a shock defeat against Portuguese qualifier Michelle Larcher De Brito.
The highest-ranked player left in that section is the 2011 champion Petra Kvitova, seeded eighth, who was trailing Ekaterina Makarova, the Russian 25th seed, 2-1 in the final set of their third round clash when play was suspended due to darkness.
Larcher De Brito was unable to capitalise on her win over Sharapova as she lost 7-5, 6-2 to Italy’s world number 104 Karin Knapp.
Knapp will face France’s 2007 runner-up Marion Bartoli who put out Italy’s Camila Giorgi, 6-4, 7-5.