The 25-year-old, who was also linked to Liverpool and Juventus, completed his move to the Emirates yesterday in a reported £30m deal.
The Chile international is determined to win silverware with the Gunners, who will compete in the Premier League, Champions League, Capital One Cup and FA Cup next season.
"Playing regularly wasn't the only reason why I came here," Sanchez told the club's official website. "I came here to win the league title, the Champions League and every title at stake.
"I feel happy to be here. I know this club has a manager that looks after his players and takes care of them as well. The most important thing is to be calm and not to rush things.
"Everything will be fine as long as I take it easy and calm down, but I am also looking forward to showing why I am here."
Sanchez spent three years at the Camp Nou, scoring 39 goals.