Andy wins on return from injury
The Briton, 29, was broken in the first game of the match but soon asserted his class en route to a 7-5 7-5 win.
The world number one last played on the ATP Tour in Indian Wells on 12 March, though he contested an exhibition match against Roger Federer on 10 April.
The British number one, handed a first-round bye, meets Albert Ramos-Vinolas next.
Murray, the top seed in Monte Carlo, gave up three double faults in the first game, then hit long to gift Luxembourg’s Muller the break.
In his first match of the year on clay, Murray continued to labour on his serve but somehow limited the damage, saving break point in the next game then fending off a set point at 3-5.
Muller’s failure to capitalise on Murray’s rustiness was then brutally exposed by the world number one, who broke the world number 28 in back-to-back games to claim the opening set, before recovering from an early break in the second to wrap up victory in one hour 55 minutes.
“It was a tough first match,” Murray told Sky Sports. “I started slow and wasn’t serving well at the start.
“I only started serving properly four, five days ago, so I knew it was going to take time but I didn’t expect to start the match serving like like that.”