The 22nd seed cast friendship aside and finished strongly with a 4-6, 6-2, 6-1 win on Rod Laver Arena and will now face Czech 25th seed Muchova, who earlier stunned world number one Ashleigh Barty.
Brady is the last woman standing among those forced into a hard 14-day quarantine after arriving in Australia.
“We’re such good friends. I know we will having many more tough battles,” she said of Pegula, whose billionaire father owns the Buffalo Bills NFL franchise.
“I felt the pressure from her. She’s such an aggressive player,” added Brady, who hit 22 winners and seven aces. “I was looking to push her back and get more on offense. I think I played a really good third set.”
Next stop: semifinals ✅@jennifurbrady95 dropped the first, recovered, then never looked back! Brady defeats Peg… https://t.co/EWgyClRiEs
— #AusOpen (@AustralianOpen) 1613535133000
Brady, who had only been broken twice in the tournament before the match, was under pressure from Pegula, who won 14 points in a row to dominate the early exchanges in humid conditions.
Brady, 25, slammed down four straight aces in the seventh game but her serve then became errant as Pegula sealed the first set.
Brady skipped out to a 3-0 lead in the second as she eyed forcing a decider but relinquished her advantage and threw her racquet in frustration after losing a point.
The outburst, much like Novak Djokovic on Tuesday, seemed to refocus Brady, who then ran away to take it to a deciding set.
The see-saw continued in the third set with breaks traded before Brady took control of Pegula’s serve to reach the final four of a Grand Slam for the second time.
Before last year’s US Open, where she lost an epic semi-final to eventual champion Naomi Osaka, Brady had been bounced out of the first round four times in her previous five majors.