The Russian world number three eventually tamed the Polish ninth seed 7-6(4), 2-6, 6-3, 5-7, 6-4 in draining heat on Rod Laver Arena after a contest lasting nearly four hours.
Playing his 100th Grand Slam match, Medvedev is now into an eighth major semi-final, but has only gone on to win only one title -at the 2021 US Open.
Medvedev made the final in Melbourne in 2021 and 2022, but succumbed to Novak Djokovic and then Rafael Nadal.
"After every match, in the locker room I'm destroyed," he said.
"But then we do a good job. One day off is probably enough to feel good the next day.
"So far, so good in the beginning of the matches, and that's what matters. Then try to win, and then if you're dead after, doesn't matter because you have a day off."
Medvedev, who will play his semi-final on Friday, admitted he was still feeling the effects of his second-round five-setter against Emil Ruusuvuori, which finished at 3:40 am last week in what has been a gruelling tournament.
He said he was the type of player who struggled in long matches and the heat.
"Sometimes I see some guys, like Hubi (Hurkacz) is one of them, and I see them play five-set matches, 7-6 in the fifth, they seem fine in the locker room," he said.
"I'm, like, 'Wow'. Either maybe it's a question of metabolism and it's genetics. I honestly have no idea. I know I get tired.
"I would love to be someone who is not really tired, doesn't care about the heat, but that's not me. But I try to win as I can."
Hurkacz has one of the biggest and best serves in the game, but Medvedev is arguably the game's top deep-court returner and he went on the attack from the outset, breaking immediately.
The Pole's nerves settled and a backhand winner from the baseline in game six brought the first set back onto level terms.
It went to a tiebreak, where too many unforced errors from the Polish player proved costly.
Angry at his lapse in concentration, Hurkacz came out firing at the start of the second set, breaking Medvedev straight away and again in the seventh game to level the match.
But a rare double fault from the Pole - only his second of the match - while 30-40 down on his opening serve in set three again gave Medvedev the upper hand.
With Hurkacz stuttering, the Russian dialled up the pressure in the fourth set with an early break, but he faltered on serve at 4-3 to let Hurkacz back and it went to a deciding set.
The critical moment of the match came when Hurkacz was serving at 3-3 in the fifth and sent a backhand long to give the Russian the break he needed to ultimately seal the win.
