The Briton, who had a first-round bye, had not played competitive tennis since injuring his elbow in March and began with three double faults as he lost his opening service game.
Left-hander Muller served for the first set but Murray dug himself out of trouble to move through.
Rafael Nadal began his quest for a 10th title in the principality but was also made to work hard by Murray's compatriot Kyle Edmund - winning 6-0 5-7 6-3.
Nadal made a dazzling start on one of his favourite courts but hard-hitting 22-year-old Edmund hit back to worry the Spaniard who was playing his 400th match on the red dust.
Top seed Murray, 29, was all at sea in the opening stages against Muller but eventually settled into his rhythm to scramble past the world number 28.
He takes on Albert Ramos-Vinolas next while Nadal, seeded four, has a tricky task against another young gun in Germany's Alexander Zverev who eased past Feliciano Lopez.
"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 that."
Third seed Stan Wawrinka beat Czech Jiri Vesely 6-2 4-6 6-2 while sixth seed Dominic Thiem of Austria lost only four first service points as he beat Robin Haase 6-3 6-2 to set up a meeting with Belgian 10th seed David Goffin.
Fifth seed Marin Cilic of Croatia destroyed France's Jeremy Chardy 6-3 6-0.
German veteran Tommy Haas's hopes of following up his first-round victory over Benoit Paire were dashed when he was beaten in three sets by Czech Tomas Berdych who plays Cilic next.
(Writing by Julien Pretot and Martyn Herman; Editing by Ed Osmond)
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