Arnautovic, along with team-mate Eljero Elia, was suspended from the first team after being caught speeding in April.
He remained in exile until the end of the season but, following the summer break, has now returned to first-team training under Robin Dutt and is hoping to repay the new coach's faith in him.
"It was a great feeling to be back with the team again," he said. "I had been away for four or five weeks and that hurt.
"The issue is behind us and I'm delighted to be back. It would be stupid if I didn't learn from it.
"It was a big mistake and we know that."
Speaking from Bremen's pre-season training camp in his home country, the Austria international said he felt revitalised by talks with Dutt and the club's director of sport Thomas Eichin.
"He (Dutt) has faith in me and that is important," added Arnautovic. "I've spoken to the coach and director of sport and cleared everything up."
Eichin confirmed that Arnautovic "has apologised to Werder" and that he knows he made a mistake.
"Now he gets a fresh chance and I have told him that he starts all over again from zero," added Eichin on his club's website.
"He will be treated by me and the coaches precisely like every other player. We are a team and there are rules which Marko also has to stick to, and there are also liberties which he will be able to take like everybody else."