Zverev will play seventh seed Casper Ruud in the last-four after holder Novak Djokovic's withdrawal from the other quarter-final with a knee injury, and the German got there after some testing moments midway through the clash.
The former US Open runner-up took control and claimed the see-sawing first set after a crucial break in the seventh game, but was dragged into a dogfight in the next after the duo swapped serves again and went to a tie-break.
Zverev fought from 4-0 down in the tiebreak and reached set point after a breathless 39-shot rally, doubling his lead in the contest shortly after and soaking up the applause from the Court Philippe Chatrier crowd.
De Minaur gifted his opponent a break and a 4-2 lead with a double fault but battled tenaciously to recover it three games later, only to surrender serve again with Zverev closing out the victory thanks to an unforced error.