Palmieri played two seasons for the USA Hockey National Team Development Program before he was selected by the Anaheim Ducks in the first round (No. 26) of the 2009 NHL Draft.

He played the 2009-10 season for the University of Notre Dame and had 17 points (nine goals, eight assists) in 33 games. He then turned pro and scored 62 goals over his first two seasons with Syracuse of the American Hockey League.

Palmieri never scored more than 14 goals over five seasons with the Ducks and was traded to the New Jersey Devils on June 27, 2015. He is a seven-time 20-goal scorer since the trade, including 30 in his first season with New Jersey (2015-16). Palmieri represented the Devils at the 2019 NHL All-Star Game; he finished the 2018-19 season with 50 points (27 goals, 23 assists).

Palmieri was traded to the New York Islanders by the Devils with Travis Zajac on April 7, 2021, for A.J. Greer, Mason Jobst, a first-round pick in the 2021 NHL Draft and a fourth-round pick in the 2022 NHL Draft. He signed a two-year contract with the Islanders on May 30, 2025, in lieu of becoming an unrestricted free agent, after his 24 goals and 48 points each ranked third on the team that season.

On the international stage, Palmieri helped the United States win the 2010 IIHF World Junior Championship with nine points (one goal, eight assists) in seven games.

The Smithtown, New York, native grew up in New Jersey. He attended high school at St. Peter's Prep near Montvale and played for the New Jersey Devils Youth Hockey Club in West Orange.