Larkin collided with Rangers forward Noah Laba at 8:39 of the second and struggled to get to the bench, where he remained and missed a few shifts before returning.
He finished with seven shots on goal in 20:45 of ice time.
"It was an awkward play, awkward fall ... a little bit of a stinger," Larkin said. "Got up and shook it off."
Less than 30 seconds after J.T. Compher hit the post on a short-handed chance in the third period, Perreault scored his second of the game for a 3-0 lead with a power-play goal at 7:13. The 20-year-old tapped in a pass from Jonny Brodzinski.
"Obviously that (goal) would have been nice, but at the same time, we shouldn't have let it got to that point," Detroit forward Lucas Raymond said. "We’ve got to find ways to put pucks in the net."
Perreault completed his hat trick with an empty-net goal at 18:16 to make it 4-0. He finished with a team-high six shots.
"I'm so happy for him. He played great today," Rangers captain J.T. Miller said.
Perron ended Quick's shutout with 33 seconds left, knocking in a loose puck at the far side of the net after a wraparound attempt by Emmitt Finnie for the 4-1 final.
"You’ve got to score more than one and you’ve got to give up less than three. Our team did neither," McLellan said. "We had some opportunities. They just don't go in, but other than that, I think we can get out of the gate quicker."
The Red Wings outshot the Rangers 18-6 in the third.
"We've got to forget about this one quick and get ready for Minnesota tomorrow, and that's the focus for me right now and for our group and we talked about it after," Larkin said. "… We can't dwell on this one -- get a little better around their net and hungrier to put pucks away."
NOTES: Perreault became the fourth Rangers rookie in the past 30 years to score a hat trick, joining Chris Kreider (Nov. 30, 2013), Derek Stepan (Oct. 9, 2010) and Jan Hlavac (Feb. 11, 2000). … Red Wings defenseman Justin Faulk (lower body) and forward Mason Appleton (upper body) were unavailable; McLellan said each is questionable to play Sunday.