Marcus Johansson, Blake Coleman, Jean-Sebastien Dea and Brian Boyle also scored for the Devils who are off to a 2-0-0 start. It was a rare victory for New Jersey in the series. Washington was 14-1-2 in the previous 17 meetings.
Pheonix Copley stopped 30 shots in his Capitals' debut.
The Devils' defense did an outstanding job stifling the Capitals, who had scored 18 goals in the first three games.
Palmieri scored twice in the first period to stake the Devils to a 2-0 lead. The Devils had the better jump at the start, outshooting the Capitals 13-6 over the first 20 minutes.
Washington, playing back-to-back after an emotional 5-2 win over Vegas on Wednesday in a rematch of the Stanley Cup final, started a step slow and New Jersey capitalized.
Palmieri put the Devils on the board at 7:02 on a shot that glanced off his skate in a goalmouth scramble.
His second goal was a power-play drive from the top of the right circle at 18:33. That gave Palmieri four goals in his first four periods of the season. It was also the 16th time he has scored at least two goals in a game.
Hopes for a Washington rally in the second were quickly dashed when Johansson scored against his former team 2:54 into the period. His shot from the blue line found the net behind a badly screened Copley.
Copley kept the Caps in the contest early in the third, stopping Taylor Hall on a short-handed breakaway.
Coleman put the game out of reach at 4-0 when he beat Copley to the short side from a sharp angle at 5:49. Dea added an insurance tally at 11:35 while Boyle scored with 5:15 left
The remaining drama centered on Kinkaid's bid for a fifth career shutout. He made an outstanding stop on Jakub Vrana, stretching out to make a pad save midway through the period.
NOTES: This was the first of four straight home games for New Jersey after opening the season with a 5-2 win over Edmonton in Sweden. . Kinkaid will carry the goaltending load as Cory Schneider continues his recovery from offseason hip surgery. Schneider has resumed skating with the team. . Hall was greeted by ''MVP'' chants as last year's Hart Trophy winner was introduced to the Prudential Center crowd. . New Jersey scratched defensemen Eric Gryba, Steve Santini and RW Drew Stafford. The Washington scratches were C Jayson Magna, D Madison Bowey, RW Tom Wilson and LW Nathan Walker.
Washington: Hosts Toronto on Saturday.
New Jersey: Hosts San Jose on Sunday.