Murray lost his serve in the opening game of the match, but came back to lead 5-4 before De Minaur won the opening set in a tiebreak.
The Briton took the second set and at 5-2 up in the decider, looked set for victory.
Despite serving twice for the match, Murray lost momentum as De Minaur's comeback saw him break his opponents serve three times winning the last five games to claim victory in the battle which lasted three hours.
Murray smashed his racquet in frustration after the agonising loss, and he has yet to win a match in this competition since winning the tournament in 2016.
Alejandro Davidovich Fokina defeated 15th seed Ben Shelton, the Spaniard overcoming the American 7-6(4) 5-7 6-3, while Karen Khachanov recorded a routine win over Max Purcell, the 16th seed Russian winning 6-4 6-4 against the Australian.
Persons:
Andy Murray, Switzerland's Leandro Riedi, Jason Cairnduff, Alex de Minaur, De Minaur, Murray, Minaur's, Frances Tiafoe, Alexander Bublik, Taylor Fritz, Argentina's Sebastian Baez, Tommy Paul, Frenchman Richard Gasquet, Alejandro Davidovich Fokina, Ben Shelton, Karen Khachanov, Max Purcell, Novak Djokovic, Holger Rune, Carlos Alcaraz, Daniil Medvedev, Trevor Stynes, Anita Kobylinska, Christian Radnedge
Organizations:
AO, Paris, ATP, Murray, Briton, Thomson
Locations:
Britain, Switzerland, Manchester, Turin, American