Rasmussen scored his 15th goal for a 1-0 lead 1:08 into the first period on a snap shot from the high slot.
Dawson Mercer tied it 1-1 for New Jersey on a backhand from the slot at 17:14, his first goal in 19 games.
Moritz Seider gave Detroit a 2-1 lead on a wrist shot from the point at 6:30 of the second period. Seider, a favorite to win the Calder Trophy as the NHL rookie of the year, finished the season with 50 points (seven goals, 43 assists).
Nolan Foote tied the game 2-2 for the Devils when he deflected a shot from the point by Hamilton at 15:38.
Joe Veleno made it 3-2 for Detroit on a snap shot from the slot 54 seconds into the third period.
Zetterlund tied it 3-3 at 6:57 on a snap shot from the left circle off a drop pass by Hischier.
"We're a young team but at some point we've got to start winning," Hischier said. "I obviously didn't lose trust in the group we have. Sometimes it hurts, but at the end, it's going to feel even better when we win. You got to go through the lows to get to the highs."
NOTES:Seider is the fifth rookie defenseman in the past 32 years to score 50 points in a season, joining Quinn Hughes (53, 2019-20), Cale Makar (50, 2019-20), Vladimir Malakhov (52, 1992-93) and Nicklas Lidstrom (60, 1991-92). ... Seider, Mercer and Detroit forward Lucas Raymond are the only NHL rookies to play in all 82 games this season. ... Hischier went 15-for-20 on face-offs (75.0 percent) and played 19:12 in his 300th NHL game. ... Zetterlund finished the season with a three-game point streak (two goals, two assists).