De Minaur keeps alive his chances of qualifying for the ATP Finals in Turin, as the 13th seed won five consecutive games in the last set in a 7-6(6), 4-6, 7-5, victory, his sixth win over Murray in their six meetings.
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.
Frances Tiafoe, 14th seed, was also in the running for the ATP Finals, but the American's dream died after suffering a straight-sets defeat to Alexander Bublik, the Russian winning 6-3, 6-4.
American ninth seed Taylor Fritz, meanwhile, advanced to the second round past Argentina's Sebastian Baez 6-1, 6-4 before fellow American Tommy Paul, seeded 12th, came from a set down to beat Frenchman Richard Gasquet 0-6, 6-2, 7-6(6).
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.
World number one Novak Djokovic, defending champion Holger Rune, Carlos Alcaraz and Daniil Medvedev are among the main seeds who received a bye into the second round.
