The 22-year-old winger was a restricted free agent after scoring 10 regular season goals and three in the playoffs last season.
Anaheim selected Palmieri with the 26th pick in the 2009 NHL Draft.
Palmieri scored 10 goals and had 21 points in 42 games during the lockout-shortened regular season.
He has 15 goals and 14 assists in 70 career games with Anaheim.
Terms of the deal were not announced but the Orange County Register said the contract was valued at $4.4 million.