The 35-year-old produced 26 unforced errors in an uncharacteristically sloppy display, and he was unable to produce a trademark comeback as he fell to a 6-1 6-3 defeat.
A slow start saw Murray lose the first four games and while he eventually got himself on the board, he was a set down within 35 minutes. He took his frustation out on the umpire early in the second, unhappy a point was not stopped after his hat fell off, but it did not turn the Brit’s fortunes around.
De Minaur, who has now won all three of his ATP Tour matches against Murray, served it out at the first time of asking to book a second-round clash against qualifier Jan-Lennard Struff.
Murray confirmed his desire earlier this year to play a significant portion of the clay swing, having been encouraged by his form and fitness in the early weeks of the season. He reached the third round of the Australian Open after a succession of marathon matches, before making the final of the Qatar Open where he was beaten by Daniil Medvedev.
Injuries have prevented Murray from regularly competing on clay in recent years, and it has not been a surface he has been entirely comfortable on during his career.
It remains to be seen when Murray next returns to the court as he builds towards the French Open. He reached at least the semi-finals at Roland Garros in four successive years between 2014 and 2017, but has played just once at the tournament since then. That was a 6-1 6-3 6-2 first-round defeat to Stan Wawrinka three years ago.
British number one Cameron Norrie was also beaten in straight sets in Monte Carlo, by Francisco Cerundolo, but Jack Draper returned from injury to battle past Sebastian Baez in a 6-3 7-5 win. He will face tenth seed Hubert Hurkacz next.