The second-seeded De Minaur wasted four successive match points at the end of the fourth set and Munar saved another on his service in the fifth but the 19-year-old booked his place in the final on his sixth opportunity.
De Minaur is now an impressive 28-22 on the year after entering 2018 with just two tour-level match wins to his record.
Stefanos Tsitsipas, ranked 15th in the world, came through 2-4, 4-1, 4-3 (7/3), 4-3 (7/3) against Alex de Minaur for his second title after Stockholm last month.
He said he didn't know how he recovered from those match points.
"It feels great. I've been playing some great tennis this week here in Milan", Tsitsipas said. It was a fantastic match, I stayed calm.
Stefanos Tsitsipas and Alex de Minaur will meet in the championship match of the Next Gen ATP Finals in Milan, after both went the distance in respective victories on Friday.
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The tournament is being played under a revised format that includes shorter sets to four games, best of five-set matches, no AD scoring, no lets and a shot clock.
De Minaur added: 'It's been an unbelievable year.
Munar scored three mini-breaks in the tie break to win it 7-5 after an ace, continuing where he left in the previous matches and setting the ground for another surprising win. Congratulations to Stef and his team for an incredible year and an unbelievable week'.
Rublev eased past Italian wild card Liam Caruana 4-3 (9-7), 4-1, 4-2 to book his passage after losing to de Minaur on Wednesday, with Munar beating American Frances Tiafoe 4-1, 4-3 (7-3), 4-1 to seal the final berth.
"As for @alexdeminaur - I am looking forward to seeing you play in Australia this summer".
'Hopefully I'm going to make the pre-season like I never did before and I can perform well next season and can compete with the top players'.