British second seed Murray survived a second set wobble to carve out a 6-4, 7-6 (7/5), 6-1 win over Russian 20th seed Mikhail Youzhny to reach the quarter-finals.
Murray, the runner-up to Roger Federer in 2012, will face Spain's Fernando Verdasco for a place in the semi-finals.
The Scot has an 8-1 winning record over unseeded Verdasco, a former top-10 player.
Murray had to battle back from 2-5 down in the second set to quell the threat posed by Youzhny, who needed treatment on a shoulder injury after one game of the third set.
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"It was a tough match, the first couple of sets especially. He also had a few chances at the beginning of the third set as well but once I got ahead in the third I concentrated very hard not to let him back in like I did in the second set," said Murray.
The Scot said he wasn't getting carried away by the national hysteria as he tries to become Britain's first champion since Fred Perry in 1936.
He is wary of becoming another shock Wimbledon victim, like Roger Federer and Rafael Nadal.
Verdasco, a former world number nine who now stands at 54 in the world and is playing his 41st successive major, reached his first Wimbledon quarter-final with a 6-4, 6-4, 6-4 win over unseeded Frenchman Kenny De Schepper.
Davis Cup teammates Janowicz and Kubot set-up the first ever all-Polish quarter-final.
Janowicz, the 24th seed, defeated Austria's Jurgen Melzer, 3-6, 7-6 (7/1), 6-4, 4-6, 6-4 on the back of 16 aces and 34 winners as he reached his first Grand Slam quarter-final.
The last Pole to reach the last-eight at Wimbledon was Wojtek Fibak in 1980.