Dimitrov seemed to be on his way to a straight set win after clinching the first two sets, but had to overcome a late revival by Kyrgios to seal the victory after three hours and 26 minutes, reports Efe.
"What can I say? Playing against Nick is always tricky. He deserves a lot of credit. He fought really hard," Dimitrov said after the win.
Both players held serve through the first set, with Dimitrov squandering three break points, but he did not offer his rival any.
They traded serve breaks, one apiece during the second set, pushing it to a tie break, which Dimitrov clinched.
Kyrgios turned up the heat in the third set, seizing on one break point out of the two he got to close the gap.
Again, each player dropped his service game once in the fourth set, which Dimitrov sealed in the tie break.
The last time an Australian won the country's Grand Slam event was in 1976, when Mark Edmondson clinched the title.
The reigning champion of the ATP Finals, Dimitrov is set to take on Britain's Kyle Edmund, who booked a place in the last-8 round in a Grand Slam for the first time, defeating Andreas Seppi of Italy 6-7 (4-7), 7-5, 6-2, 6-3.
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