The NHL announced the deal late Friday as the first round of the draft was coming to an end in Sunrise, Florida.
New Jersey shipped the 41st overall pick in the draft and a third-round choice next year to Anaheim for Palmieri. The Devils acquired the 41st pick earlier this year in a deal that sent Jaromir Jagr to Florida.
The 24-year-old Palmieri had 14 goals in 57 games with the Ducks last season. He was born in Smithtown, New York, grew up in Montvale, New Jersey, and played collegiately at Notre Dame.