The world number 81 beat Thailand's Luksika Kumkhum 6-3 3-6 7-5 to set up a clash with unseeded Belgian Elise Mertens.
Twelve months ago Martic was ranked 264 after a back injury but she has now reached the last 16 of three of the last four grand slams and hopes to go even further.
"I think at every grand slam it opens up sooner or later. I think the competition is so high, anybody can beat anybody. I think anybody can win this tournament.
"I think it's great to see, because it's very exciting, and, you know, you can't say who's gonna win the tournament, who's gonna win the semis.
"It's pretty open, and I think I'm one of those girls that are kind of not expected to be here but am still there."
(Reporting by Martyn Herman; Editing by Amlan Chakraborty)
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