New Jersey received forwards A.J. Greer and Mason Jobst, a first-round pick in the 2021 NHL Draft and a fourth-round pick in the 2022 NHL Draft.
Palmieri and Zajac can each become an unrestricted free agent after this season.
Palmieri scored 17 points (eight goals, nine assists) in 34 games this season and scored at least 24 goals in each of the prior five seasons. The 30-year-old forward is in the final season of a five-year contract he signed with the Devils on July 7, 2016.
"The opportunity to go and be a part of a team that has a chance to do something special is something every player wants," Palmieri said. "That's all you can ask for. I'm really looking forward to this opportunity."
Zajac scored 18 points (seven goals, 11 assists) in 33 games this season. The 35-year-old center is in the final season of an eight-year contract he signed with the Devils on Jan. 16, 2014.
Selected by New Jersey in the first round (No. 20) of the 2004 NHL Draft, Zajac scored 550 points (202 goals, 348 assists) in 1,024 regular-season games for the Devils, and 28 points (11 goals, 17 assists) in 57 playoff games.
His games played are fourth in New Jersey history, behind Ken Daneyko (1,283), Martin Brodeur (1,259) and Patrik Elias (1,240).
"I'm extremely excited for this opportunity," Zajac said. "At [this] point in my career and where I'm at, I want a chance to win. I just believe this is a team that has all the aspects of a winning hockey team and to be able to try and be a part of that and help them in any way possible ... I think it was just, for me, it was the right time.
"I was fortunate enough to have a great career in New Jersey and a lot of great memories, a lot of great people along the way, players, coaches, staff. But I think it just came down to, at this point I want a chance to try and reach that ultimate goal."
[RELATED: 2020-21 NHL Trade Tracker | Fantasy spin: Palmieri to Islanders]
The Devils traded defenseman Andy Greene to the Islanders on Feb. 16, 2020, for a second-round pick in the 2021 draft and defenseman prospect David Quenneville. Greene had played his entire 14-season NHL career with the Devils and was captain for five seasons. He played 923 regular-season games for New Jersey, seventh in its history.
"Obviously, [it's] very exciting," Greene said. "Two great guys, great people, great players. It's a good message for our team, and now it's on us. We've got to continue to do what we've been doing up to this point and continue to build and be ready for the home stretch, and then after that.
"As people and players, they should fit right in. The way they play, their style, what they bring to the table both individually should be a really big help for our team."
Zajac said he spoke with Greene after the trade was completed.
"He speaks highly of the team, of the coaches, of the staff, the players," Zajac said. "Obviously, you see the success he's had this last year and how he's playing, so it's exciting to get back in action with him."
Islanders general manager Lou Lamoriello was Devils general manager from 1987-2015.
"I know what they bring on the ice," Lamoriello said. "I also know who they are as people and how well they'll fit in the locker room here. Chemistry is extremely important to me, but their experience, their talents, their competitiveness, especially in this division, knowing what they have to go through, to me, is an excitement from our end of it in management, coaching and so forth."
The Islanders had NHL salary cap space become available after forward Anders Lee injured his ACL in a game against the Devils on March 11 and was placed on long-term injured reserve. Lee, who is Islanders captain, was playing on New York's top line with Mathew Barzal and Jordan Eberle and scored 19 points (12 goals, seven assists) in 27 games. He had surgery the week of March 15 and is out for the season.
New York (25-10-4) won eight of 12 games without Lee and were tied for first place with the Washington Capitals in the MassMutual East Division.
"When we lost Anders, we didn't know what to expect because of what he brings both on and off the ice," Lamoriello said. "He's still a part of this group. He's here every day and he's still our captain and still our leader. The way the team responded to that and the way they came together, not that they haven't been together, it certainly played a role in whatever I could do to get them some support and some help."
Video: The Islanders acquire Palmieri and Zajac
Lamoriello did not rule out making another move before the NHL Trade Deadline on April 12. New York reached the Eastern Conference Final last season, losing to the Tampa Bay Lightning in six games.
"If there's a chance to get better, you have to do that," Lamoriello said. "And then, at what price is it? [But] we're pretty much content with what we have and who we have right now. Extremely comfortable."
The Devils (13-18-6) were seventh in the eight-team East, 14 points behind the Boston Bruins for fourth place with 19 games remaining. The top four teams will qualify for the Stanley Cup Playoffs.
New Jersey general manager Tom Fitzgerald said the Devils attempted to re-sign Palmieri but then decided to trade him.
"Like any negotiation, there are some differences and [it got] to the point where I felt, because of the amount of interest that teams have shown Kyle, [it] was probably time appropriate for the organization to listen and see what Kyle's value was out there as a [free agent] rental," Fitzgerald said. "Kyle and I, we have a great relationship, we got to the point on Saturday, we got together for a couple hours and just kind of talked about his future, talked about his past, my experiences and potential trades and deadlines. ... I just told him I have to do the best for the organization here and try to make the best deal possible."
Zajac had a no-movement clause and needed to agree to the trade. He is third in goals, third in points and fifth in assists in Devils history.
"From the first day ... he got to New Jersey to today, his practice today, he's just a true professional," Fitzgerald said. "True gentlemen. He'll always be a New Jersey Devil, and the value he still brought to the organization was very high, but he looked at it, if I can have one kick at it, this is it."
Lamoriello was athletic director at Providence College when Fitzgerald was a freshman there.
"I feel my relationship with Lou is extremely strong, there's a trust factor between the both of us, and that's important to Lou and it's very important to me," Fitzgerald said. "Last year we were dealing with the same thing ... there are players that Lou values on our team, for obvious reasons. For me, players want to play for Lou, especially his ex-players, and I think that's important through all this, and Travis was no different. You know he knows what the environment's going to be, he knows what the expectations will be, what he will play under, and he wanted that. No different than Andy, it's these players have earned the right to choose where they want to go, and I was I was just happy to do that for them."
Greer scored six points in 37 NHL games with the Colorado Avalanche. The 24-year-old had one goal and one assist in 10 games with Bridgeport of the American Hockey League this season.
Jobst has not played in the NHL and the 27-year-old had two assists for Bridgeport.
Palmieri was selected by the Anaheim Ducks in the first round (No. 26) of the 2009 NHL Draft. He scored 355 points (183 goals, 172 assists) in 595 regular-season games with the Devils and Ducks, and 15 points (eight goals, seven assists) in 38 Stanley Cup Playoff games.
He was born in Smithtown, New York and owns a home on Long Island.
"When I kind of heard that the Islanders could be an option, I couldn't have been more excited," Palmieri said. "As the days kind of wore on here, there was a little bit of anxiety as far as waiting for that call. But when I got that call tonight, it was really difficult to contain the excitement to join this team and this organization."
NHL.com staff writer Mike G. Morreale contributed to this report