The Brit swept the big-serving lefty aside, 7-5, 7-5, on Court Rainier III and will look forward to his last 16 clash with the Spaniard Albert Ramos-Vinolas on Thursday.
Murray has featured only once – in a charity exhibition with Roger Federer last week -- since his shock round two loss to qualifier Vasek Pospisil in Indian Wells on March 12 after he was diagnosed with an elbow ailment that ruled him out of the Miami Open.
But the 29-year-old said he did not experience any extra discomfort in his elbow during his victory over Muller.
Little to worry about
“My elbow's always a bit sore, that's nothing to do with this week or the last two years,” said Murray.
“The first service game was obviously not ideal, but when you are coming back from not serving for a few weeks that's normal.”
The extended lay-off gave Murray time to recuperate and spend time with his one-year-old daughter, Sophia, ahead of a frenetic clay court swing.
“Missing Miami was a shame because it's one of my favorite ones, but I needed it,” he added.
“Sometimes having a break, if you're feeling tired or fatigued is great.
“But when you're playing really well and winning, you don't think so much about fatigue.”
Elsewhere in Monte Carlo, Rafael Nadal was pushed to three sets by Murray’s Davis Cup teammate, Kyle Edmund, but prevailed 6-0, 5-7, 6-3.
The wily German Jan-Lennard Struff recorded the upset of the day when he ousted Australian Open semifinalist Grigor Dimitrov, 4-6, 6-3, 6-2.
Meanwhile, the Argentine dirt-baller Diego Schwartzman also notched a notable upset when he battled past Roberto Bautista-Agut 6-3 7-6(3).