The world number one was joined in the third round by nine-time champion Rafael Nadal, who held off British surprise-package Kyle Edmund 6-0, 5-7, 6-3 in the Spaniard's 400th career match on clay.
Murray waited until late in both sets to make his move, finally putting Muller away in just under two hours as he builds up to the French Open in May-June.
But the 29-year-old was poor on his serve with eight double faults while breaking Luxembourg's Muller four times.
"I definitely played better as the match went on, obviously first service game was not ideal," Murray said.
"But when you are coming back from not really serving for a few weeks, it's normal.
"Maybe the technique changed a little bit. You get into the match, and it's tricky."
Murray was kept on the back foot in the opening set as he lost the first game and only caught up late.
In the second he was also far from his fluent best, before grinding out victory.
It was the first time back on the ATP since his elbow problem -- his last match was a second-round loss to Vasek Pospisil at Indian Wells on March 12.
The three-time Monte Carlo semi-finalist Murray, who has had a sticky start to 2017 with form and injury, now stands 13-3 on the season and has one title from Dubai in his pocket.
Murray said: "My elbow's always a bit sore, always. That's nothing to do with this week.
"The last two years, always when I serve, it's just a little bit sore. But it was nothing out of the ordinary today."
Victory for Nadal took his record on clay to an outstanding 366-34.
The Spaniard swept the first set 6-0 against 45th-ranked Edmund, but found himself fighting for survival in the third set.
The fourth seed finally squeezed home after two and a quarter hours.
"What I feel today is not about confidence, is about I have the chance to play again tomorrow," Nadal said. "That's the most important thing for me.
"I saved a tough match against a player who played so aggressive and very well. Tomorrow I have another very tough battle against a big player again (Alexander Zverev)."
Murray and Nadal were joined in round three by Swiss third seed Stan Wawrinka, who saw off Czech Jiri Vesely 6-2, 4-6, 6-2.
Three-time Grand Slam winner Wawrinka won Monte Carlo three years ago over compatriot and friend Roger Federer.
"Overall it was a good match, the first match on clay," said the Swiss. "The feeling was good. Winning in three sets means it was a good victory. It's important to win and keep going.
"I prepared well, practise was good. Physically I'm fine.
"I know my level is good. If I'm able to improve I might have a chance in this tournament."
Czech ninth seed Tomas Berdych put an end to the run of 39-year-old Tommy Haas 3-6, 6-1, 6-4.
German qualifier Jan-Lennard Struff stunned Bulgarian eighth seed Grigor Dimitrov 4-6, 6-3, 6-2.