Rafa Nadal and Maria Sharapova reached the last eight and local boy Nick Kyrgios also getting through after beating Andreas Seppi to face Murray in the quarters.
Murray beat Dimitrov 6-4 6-7(5) 6-3 7-5, with Dimitrov blowing a 5-2 lead in the fourth set to lose.
"I think I got quite lucky at the end, a few net cords went my way, that was the difference really," Murray told reporters. "The momentum was switching both ways all the time."
Third seed Nadal, whose 2014 season was cut short due to injuries, was in great form beating South African Kevin Anderson 7-5 6-1 6-4.
The Spaniard now faces Tomas Berdych, who he has never lost against, for a place in the semi-finals.
"I am one of the eight. That's the most important thing," said Nadal.
"It would be very arrogant if I say I'm not at a very high percentage… I probably played my best match of the year."
Sharapova easily hammered U.S. Open semi-finalist Peng Shuai 6-3, 6-0 and will now face Canadian Eugenie Bouchard who beat Romanian Irina-Camelia Begu 6-1 5-7 6-2.
Bouchard gave herself a pep talk during a toilet break after suffering a mid-match meltdown.
"I gave myself a good long hard look in the mirror and I said, 'Genie, this is unacceptable' and I really kind of kicked myself in the butt a little bit," said Bouchard.
Australian youngster Kyrgios blew a 4-1 lead in the final set of his close 5-7 4-6 6-3 7-6(5) 8-6 win against Italy’s Seppi, to wild celebrations from the Aussie locals.
Seventh seed Berdych was all class as he beat Australia’s Bernard Tomic 6-2 7-6(3) 6-2.
Russian left-hander Ekaterina Makarova reached her third Melbourne Park quarter-final with a 6-3 6-2 thumping of German Julia Goerges.
The 26-year-old should provide a big challenge for third seed Simona Halep who cruised past Yanina Wickmayer 6-4 6-2.