Murray looked set to be dragged into a decider when Dimitrov led 5-2 in the fourth set, but won the next five games to complete a 6-4 6-7 (5/7) 6-3 7-5 success in just over three and a half hours.
The match started on Sunday evening but fittingly stretched into the first hour of Australia Day (January 26) and Murray will now look to improve on his perfect 10-0 record against Australian players at tour level when he takes on home favourite Nick Kyrgios for a place in the semi-finals.
The 19-year-old had earlier come from two sets down to beat Andreas Seppi, who caused the shock of the tournament by knocking out Roger Federer in the third round, 8-6 in the fifth set.
Murray weathered an early storm from world number 11 Dimitrov, who raced into a 3-0 lead before Murray won six of the next seven games to claim the opening set.
That set the pattern for an unpredictable contest with Murray twice going a break up in the second set and serving for it at 6-5, only to crack under the pressure and serve a double fault to allow Dimitrov to get back on level terms.
Trailing 3-2 in the tie-break, Murray then failed to put away an overhead at full stretch to allow Dimitrov a backhand winner down the line as the 23-year-old moved 6-3 in front.
Murray saved two points on his own serve but could do nothing about the third as a service winner gave Dimitrov the set in 70 minutes and levelled the match at one set all.
Murray must have been furious to have been pegged back and his mood was not improved in the third set when he was unable to take any of the four break points he created in the fourth game.
The object of his anger then switched to umpire Jake Garner, who failed to spot the ball had bounced twice before Dimitrov retrieved it.
Murray briefly stopped, thinking the point was over, but thankfully reacted quickly enough to put away a winner, before letting Garner know his feelings in no uncertain terms.
Garner may therefore have been quietly pleased when Murray finally broke Dimitrov in the eighth game and this time had no trouble serving out to take the set 6-3 and go two sets to one up.
Dimitrov, who put Murray out of Wimbledon last year, had twice come from a set down to beat Marcos Baghdatis in the third round and was not about to throw in the towel, saving yet another break point at the start of the fourth set.
Murray was understandably frustrated that he had converted just five of his 16 break points and Dimitrov compounded the agony by taking his fourth break point out of seven to surge into a 3-0 lead.
Dimitrov had to work hard to hold onto his next two service games and it was beyond midnight when Murray saved a set point to force his opponent to serve for the set.
And that proved crucial as the 10th seed committed two errors on his previously reliable forehand to allow Murray back into the match, a chance the former Wimbledon champion gratefully accepted.
After holding serve to level at 5-5, Murray broke Dimitrov again with a backhand return winner that had the Bulgarian smashing his racket into the ground before standing on it to ensure he had broken it in half.
Murray still had to complete the job and in the end was grateful for the assistance of a massive net cord to seal a thrilling win.
“I did get quite lucky in the end, a few net cords went my way and that was the difference,” Murray said in an on-court interview.
“A lot of the games were close with the momentum switching back and forth and once I got it I just tried to keep it going.”