The Austria international forward joined the Potters on transfer deadline day from Werder Bremen for a reported transfer fee of £2m.
"The trainer gave me a very good feeling about the club when he came over to Germany to speak to me," Arnautovic told The Sentinel.
"He said many good things and made Stoke City a very good option for me. I am really happy to be here and I have to thank the club for giving me this opportunity to come here and play in the Premier League.
"I hope that I can pay them back for the faith they have in me with good performances."
Arnautovic could make his debut for Stoke against Manchester City in the Premier League on Saturday.