The 25th seed looked out of sorts in a first set which lasted only 24 minutes and at this stage it seemed the home favourite sealing her place in the last four was a formality.
A medical time-out proved the catalyst for the Czech to wrestle back some momentum and, after she levelled, it was Barty who lost her radar and a catalogue of unforced errors saw hopes of a maiden Australian Open victory ended.
For Muchova her remarkable tournament continues with Karolina Pliskova and Elise Mertens already dispatched but this victory tops the lot and seals a first ever appearance at the last four of a grand slam.
In her on-court interview, the 24-year-old semi-finalist reflected on her medical time-out and said: "I started to feel a bit lost at the end of the first set.
"My head was spinning so I took a break and it helped me. I tried to get back and play a bit faster and it worked well.
"They checked my (blood) pressure, I was a bit lost and spinning. They cooled me down with ice and it helped me."
French Open winner Barty refused to be drawn into a debate about the long break in play when she attended her press conference.
She insisted: "If she wasn't within the rules, the physios and the doctors would have said so. That's the laws of our game, is that we have those medical time-outs for cases that are needed. Obviously she needed that today. Completely within the rules for her to take that.
"From my point of view, I've played a lot of matches where there have been medical timeouts. I've taken medical timeouts myself before, so that shouldn't be a massive turning point in the match.
"I was disappointed that I let that become a turning point. I'm experienced enough now to be able to deal with that."
Barty had got the better of the Czech in their previous meeting at the US Open in 2018 and signalled her intentions with a break in her opponent's first game in temperatures of 25 degrees in Melbourne.
Muchova could not find her range and a string of wayward shots were followed by a double fault to put her on the verge of a bagel following 16 minutes of play at the Rod Laver Arena.
It was avoided but Barty secured a routine first set by a 6-1 scoreline before the tide turned in unexpected fashion following a medical time-out.
The world number 27, in only her second ever grand slam quarter-final, went off for a long period after a rare winner and when she returned it was now her Australian opponent who could not land any shots.
Five of the next six games in the set went the way of Muchova to level proceedings with Barty hitting 18 unforced errors.
Momentum was firmly with the underdog and another break at the start of the decider increased the pressure on the home favourite, who was seeing hopes of becoming the first Aussie woman since Wendy Turnbull in 1980 to reach the final of the Australian Open fade fast.
With the crowd not due to return until Thursday after a five-day lockdown for the state of Victoria ends, Barty had to find support from within and yet let another game slip despite nerves getting the better of the duo with the finish line in sight.
Serving for the match, Muchova sealed her passage into the semi-finals with an ace to claim one of the best victories of her career.
The Fed Cup team-mates produced an entertaining contest with breaks aplenty in addition to various long rallies.
Brady, who reached the US Open semi-finals in August, recovered from losing the first set and being broken in the first game of the decider to fight back and knock out her good friend.
She said: "I know we will be having a lot more tough battles. Some may go her way, some may go mine. Luckily today it went mine and I am really happy to be moving onto the semi-finals."
World number 61 Pegula's fantastic journey in the tournament came to its conclusion despite a strong start where she won 12 consecutive points to seal the opener.
After 22nd seed Brady began to channel her growing frustration, winners became more frequent and three breaks secured the second set.
When Pegula broke at the start of the third, the balance of the tie looked set to turn once more, but the player ranked 24 in the world responded in style to triumph in the next six games and set up a semi-final date with 25th seed Muchova.
The winner will make a grand slam final for the first time where they will meet one of either Naomi Osaka or Serena Williams.