The relentless Russian will contest his first final at Melbourne Park on Sunday and look to claim his maiden Grand Slam title after beating Tsitsipas 6-4 6-2 7-5 at a floodlit Rod Laver Arena.
Yet Medvedev has now won 20 successive matches and 12 against top-10 opponents, including beating Djokovic on the way to winning the ATP Finals.
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.
The Serbian world number one hit back strongly after losing a tight opening set on a tiebreak but trailed in both the third and fourth sets before turning them around.
Germany's Zverev will rue allowing Djokovic back from 4-1 down in the third set and he was also up 3-0 in the fourth.
It was a patchy performance from the world number one, who is nursing an abdominal injury and drew deeply on his mental fortitude to win 6-7 (6/8), 6-2, 6-4, 7-6 (8/6).
The surprise of the men's draw has been Russian qualifier Aslan Karatsev, who will be looking for a semi-final place in his first Grand Slam tournament when he faces Bulgaria's 18th seed Grigor Dimitrov.
"I mean, to be honest, like, completely honest, it's absolutely ridiculous that at a Grand Slam match we're asked to leave the court for 10 minutes in the middle of the match, like, in the middle of the fourth set."
I understand the fact that Victoria is going back into lockdown and people have to go. If that's the case.
The eight-time champion looked in serious trouble when an injury sustained to his right side at 1-1 in the third set.
I don't have a great experience with tears in terms of continuing so it's in the clouds whether I step out on the court in two days, hopefully God willing I can play.
Djokovic was given a stern examination by America's Frances Tiafoe and dropped his first set of the tournament before recovering to win 6-3, 6-7 (3/7), 7-6 (7/2), 6-3.
American Williams has been stuck on 23 majors since triumphing at Melbourne Park in 2017, losing four Grand Slam finals since to remain one short of Margaret Court's record.
I felt really comfortable in the tournament leading up to this just a couple days ago when I was playing.
Yet, for once, Djokovic may hear the rumbling of hooves as the likes of Austrian Dominic Thiem and Germany's Alexander Zverev look to storm the fortress.
Djokovic won his eighth Australian Open title last year and finished the COVID-19 pandemic-blighted 2020 season with the top ranking for a sixth year.
After a year of playing in front of mostly empty stadiums due to Covid-19, he is also looking forward to some atmosphere, with up to 30,000 spectators allowed each day at the Australian Open.
Djokovic earlier told reporters he had taken a walk in a park to celebrate his release from the 14 days of quarantine players coming from overseas were obliged to undergo to play in Australia.