Cleary finished off the Kings by capitalizing on an opportunity created by Luke Glendening, who knocked Jake Muzzin to the ice and fed Cleary for his fourth goal with 2:34 to play.
Los Angeles lost in regulation for the first time in four games on a five-game homestand.
Quick hadn't lost in regulation since returning from a 24-game absence with a groin injury, but his teammates returned to their low-scoring ways after a four-goal outburst against Boston two nights ago. Los Angeles has scored more than three goals in a game just once since Dec. 19.
"You can't win every game 1-0 or 2-1," Kings forward Justin Williams said. "Certainly we had enough chances to win the game tonight, but we didn't do it. The onus is on us to put the puck in the back of the net."
After Doughty's goal late in the first period, Detroit evened it when Sheahan flipped home a puck in front while being checked to the ice, scoring his first goal in seven career NHL games over two seasons.
The 22-year-old Sheahan was a first-round pick in 2010, playing three seasons at Notre Dame before winning a Calder Cup with the Wings' AHL affiliate in Grand Rapids last year. Detroit recalled him from the AHL on Friday.
"I wasn't able to really feel the feeling when I scored," Sheahan said. "But now I'm pretty excited to go talk to my parents about it."
NOTES: Detroit also won without veteran forwards Pavel Datsyuk and Daniel Alfredsson. Datsyuk missed his third straight game with a lower-body injury, while Alfredsson sat with back spasms. ... Kings F Jeff Carter played his 600th NHL game. ... The teams meet again in a week at Joe Louis Arena.