The world number one, playing his first competitive match since early March, was understandably rusty as he came through 7-5 7-5.
He will face Spain’s 15th seed Albert Ramos-Vinolas in the third round.
"It was a tough first match, I started the match pretty slow, I obviously wasn't serving that well at the start," Murray told Sky Sports.
"I haven't been in the best rhythm on serve because I only started serving at full speed four or five days ago. I knew it would take a bit of time but I wasn't expecting to start serving like that.
"It got a little bit better, I started to create a lot of chances in the second set. I couldn't get the breakthrough until right at the end."
Murray’s comeback started in nightmare fashion as Muller broke him immediately before building a 5-3 advantage.
But the Brit soon found his groove, winning the next four games to steal the opener as his frustrated opponent slammed his racquet into the clay.
It was similarly tough in the second set before Murray made the decisive breakthrough in the 12th game to wrap up victory in 115 minutes.
Third seed Stan Wawrinka also progresed with a 6-2 4-6 6-2 win over Jiri Vesely.
The three-time Grand Slam champion looked imperious in the first set, but dropped his level in the next as his Czech rival levelled the match.
It was only a brief blip, however, as Wawrinka raced through the decider to book a match with Uruguay’s Pablo Cuevas.
Rafael Nadal is in action against Britain's Kyle Edmund later on Wednesday.
Read the original article on Eurosport: Andy Murray returns with win in Monte Carlo