Growing up on Long Island, Palmieri played youth hockey for the Devils-sponsored teams at Codey Arena in West Orange, New Jersey.

He attended St. Peter's Preparatory School for two years before earning an opportunity to play for the USA Hockey National Team Development Program (NTDP).

After being released from the NTDP, Palmieri played one season of college hockey for the University of Notre Dame Fighting Irish.

Palmieri spent one season with the Fighting Irish before turning professional with the Ducks' American Hockey League (AHL) affiliate, the Syracuse Crunch.

[5] He began playing with the Kodiak Hockey Club at McKay Ice Arena under coach Gary Hess.

[9] Upon joining the NTDP Under-18 team, Palmieri tallied 15 goals and 15 assists through 33 games and recorded a hat-trick in the gold medal round of the 2008 Five Nations Tournament.

Palmieri made his NHL debut on November 3, 2010, against the Tampa Bay Lightning and he scored his first career goal to lead the Ducks to an overtime win.

[20] He was returned to the AHL to begin the season, where he played on their top line alongside Nick Bonino and Patrick Maroon.

[27] Palmieri finished the AHL regular season with 33 goals and 25 assists through 51 games as the Crunch qualified for the 2012 Calder Cup playoffs.

While playing with the Admirals, Palmieri tallied 13 goals and 12 assists for 25 points[29] before being promoted to the NHL level once the lockout ended.

[2] Following an injury to Dustin Penner mid-game, Palmieri was placed on the Ducks' top line alongside Corey Perry and Ryan Getzlaf.

[34] On October 30, Palmieri recorded his fourth multi-goal game to help the Ducks come back from a 2–0 deficit against the Philadelphia Flyers.

[2] In the second year of his new contract, Palmieri missed the majority of the Ducks' pre-season games due to a high ankle sprain.

[44] On June 26, 2015, Palmieri was traded by the Ducks to his hometown club, the New Jersey Devils, in exchange for second- and third-round picks in the 2015 NHL Entry Draft.

[52] Upon returning to the Devils' lineup after missing one game, Palmieri slid into his previous spot on the top line alongside Taylor Hall and Pavel Zacha.

[57] During a 4–3 win over the Carolina Hurricanes on March 27, Palmieri tallied his 10th multi-point game of the season and led the club in power play goals with 11.

After Hall stepped down from the 2019 National Hockey League All-Star Game due to an injury, Palmieri was named his replacement.

[67] He subsequently returned to the lineup on March 8 and played on a line with Travis Zajac and Kenny Agostino for their game against the Capitals.

[70] By December, Palmieri led all Devils players in scoring with 27 points in 38 games,[71] while playing alongside Jesper Bratt and Nico Hischier.

[76] On January 31, Palmieri was placed on the NHL's COVID protocol list and missed a matinee game on the road against the Buffalo Sabres.

[77] As numerous other teammates tested positive for COVID, the Devils entered a 14-day pause to comply with NHL COVID-19 protocols before returning on February 16.

[80] In the Islanders' final regular season game on May 8, Palmieri and teammate Mathew Barzal tallied a goal each 37 seconds apart in the third period to win 5–1 over the Devils.

[82] During Game 1 of the 2021 Stanley Cup playoffs against the Pittsburgh Penguins, Palmieri played on the right-wing of Jean-Gabriel Pageau and rookie Oliver Wahlstrom.

[11] In the same year, Palmieri also represented his home country at the 2008 IIHF World U18 Championships where he won a bronze medal.

[95] Palmieri next represented Team USA at the 2010 World Junior Ice Hockey Championships during his freshman year at Notre Dame.

Palmieri subsequently became the first Notre Dame player to capture a gold medal in World Junior Championship play.

[98] Palmieri made his debut with the United States men's national ice hockey team at the 2012 IIHF World Championship, where he tallied two goals in seven games.

[100] As a result of this, Palmieri launched Squad21 to offer active or veteran military service members the chance to attend a Devils home game during the 2016–17 regular season.

Palmieri subsequently became the Devil's 2018 nominee for the King Clancy Memorial Trophy as a player "who best exemplifies leadership qualities on and off the ice and has made a noteworthy humanitarian contribution in his community.

The complaint contended that the city made out-of-state athletes pay a fee equal to 3% of their taxable earned income for the days in which they play games in publicly funded sports facilities.