The Sharks won in Los Angeles despite trailing after two periods for the third time in their history (3-28-2).
San Jose tied the game 1-1 when Karlsson scored his third goal of the season at 5:42 of the third period. After Brown whiffed on a chance to clear the puck, Tomas Hertl tried to sneak a shot in between two defenders and the puck got to Karlsson in the slot with an open net. Quick was unable to recover in time.
"We just pulled back and we weren't attacking as much as we did in the first couple of periods," Kings captain Anze Kopitar said. "They were able to capitalize on that, but it was just a couple of fluky goals, really."
Brown gave the Kings a 1-0 lead with a power-play goal at 8:30 of the first period when he was left all alone just outside the crease and redirected Kopitar's shot from the right circle past Jones.
Brown has eight goals in 17 games after needing 53 games to reach that total last season.
"I thought we needed to play 60 minutes tonight," Kings coach John Stevens said. "I thought we played 20 really well and then 40 the way they wanted to play."