Sixth seed Murray beat the 10th seeded Bulgarian 6-4, 6-7 (5/7), 6-3, 7-5 and will now play Australian teenager Nick Krygios in the last eight.
The British star showed fantastic fighting qualities to claw back from 2-5 down in the fourth set to deny Dimitrov taking the epic match into a fifth and deciding set.
Murray, a two-time Grand Slam champion but luckless in Melbourne in three losing finals, will now have to battle a home crowd when he takes on Kyrgios in the last eight on Tuesday.
"I got quite lucky at the end, a couple of net cords went my way and that was the difference really," Murray said of the gruelling 3hr 32min match on Rod Laver Arena.
Also Read
Murray won match point on a net cord.
"A lot of the games were close, back and forth, and the momentum was switching both ways all the time," he said.
"Grigor is an unbelievable athlete, he's extremely quick around the court, so agile, and he gets shots that other players don't get to and he's got fantastic hands as well.