The league announced the deal during the draft's second round on Saturday morning. Hagelin, 26, is a restricted free agent and will be able to negotiate with Anaheim. He totaled 17 goals and 18 assists and was plus-18 for New York in 2014-15.
Etem, 23, is also a restricted free agent. A first-round pick in 2010, he posted 10 points in 45 games for the Ducks last season.
New York selected forward Ryan Gropp with the 41st pick. Anaheim took forward Julius Nattinen at No. 59.
- Jeremy Woo