0825 GMT June 22, 2018
The world number one and top seed waited until late in both sets to make his move, finally putting Muller away in just under two hours to reach the third round, AFP wrote.
But the 29-year-old Scot was poor on his serve with eight double faults while breaking Luxembourg's Muller four times.
And Murray was 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 an elbow injury – his last match was a second-round loss to Vasek Pospisil at Indian Wells on March 12.
The three-time Monte Carlo semifinalist, 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 was 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.
Austrian Dominic Thiem, seeded sixth, advanced over Dutchman Robin Haase 6-3, 6-2 while Czech ninth seed Tomas Berdych put an end to the run of 39-year-old Tommy Haas 3-6, 6-1, 6-4.
Pablo Carreno Busta, Alexander Zverev and Albert Ramos-Vinolas all advanced.