Chasing his third claycourt title of the season, the Austrian looked in fine form in Hamburg as he claimed a decisive break in the seventh game against Moutet to win the opening set.
Moutet, ranked 119 in the world, threatened a comeback when he secured an early break in the second set, but Thiem reeled off six consecutive games to seal victory.
Earlier in the day, Frenchman Gael Monfils moved into the second round with hard-earned 6-4 3-6 6-4 win over Italian Marco Cecchinato.
Monfils will next meet defending champion Leonardo Mayer, who recorded a 6-3 6-2 victory over Spain's Albert Ramos-Vinolas.
(Reporting by Hardik Vyas in Bengaluru; Editing by Toby Davis)
