Brown tipped in a centering pass by Anze Kopitar to tie it 1-1 at 6:55. Brown ended a season-long five-game goal drought.
"They were digging in and they had a hard, hard push when the puck dropped at the beginning of the night, and I thought it took us a while to get our legs going," Sharks coach Bob Boughner said. "It's a period like that, we come out 1-1 and we got outshot badly and are still in position to change the game, and we did that in the second period."
Kane gave the Sharks a 2-1 lead at 15:04 of the second, stretching around Quick to score into an open net.
Moore tied it 2-2 at 19:06, taking the puck from Kane in the neutral zone before tapping in his own rebound from close range for his second shorthanded goal of the season.
Moore is one of 12 NHL players with multiple shorthanded goals this season. Columbus Blue Jackets forward Cam Atkinson leads the NHL with four.
"On the penalty kill, you know if you're on it, first and foremost, you've got to get your clears and you've got to be solid defensively," Moore said. "The offensive stuff is great when it happens. You only got to do it when it presents itself. You can't be forcing it, for sure."
NOTES:Sharks defenseman Marc-Edouard Vlasic did not play the final two periods because of an upper-body injury. Boughner said Vlasic would be evaluated in San Jose on Sunday. … Jones is 5-1-0 with a 2.37 goals-against average and .932 save percentage against the Kings this season. … Jones is 18-5-3 in his NHL career against the Kings. … Sharks defenseman Nikolai Knyzhov had an assist, his third straight game with a point (one goal, two assists). … Brown has scored six points (three goals, three assists) in six games against the Sharks this season.