The 26-year-old arrived at Melbourne Park having never made it past the second round of a major.
But world No.69 Borges' 6-7 (7-3) 6-4 6-2 7-6 (8-6) upset of the Bulgarian on Saturday on Kia Arena moves him through to the last 16.
"I don't think I've ever played in front of so many people. What a win," Borges said.
"I've got to thank my team, and all the Portuguese people out there bringing in the flags, it's been awesome.
"Tennis gives you confidence, but it doesn't really last long, you've got to be ready for the next one."
Borges will face third seed Daniil Medvedev as the resurgent Russian seeks to avenge his brutal loss in the 2022 Australian Open final to Rafael Nadal.
Medvedev admitted after his win over Felix Auger Aliassime that he knew little about his fourth-round opponent.
"I saw the first set today, he (Borges) was playing well," the Russian said.
"I didn't know him before, but he was playing well, and to beat Grigor, who was on fire lately, is a big achievement."
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