Doughty put the Kings ahead 1-0 from the high slot during a power play at 7:20 of the third. Malone tied it at 1 on a rebound wraparound goal with 8:21 left in the period.
Tampa Bay had a goal with 1:40 left in overtime disallowed because of goaltender interference. It appeared that Tampa Bay forward Paul Szczechura might have been pushed into Quick by a Kings player during Andrej Meszaros' shot from the blue line.
"I was going to the net to try and get into the goalie's eyes there with a screen," Szczechura said. "I got pushed in there."
Tampa Bay coach Rick Tocchet believed it should have been a goal.
"There's nothing you can do. It happens," Tocchet said. "We had a couple chances in overtime, but it's a tough one."
The call by referee Frederick L'Ecuyer wasn't allowed to be reviewed by video replay.
"In the eyes of the referee, the goaltender was clearly interfered with by the attacking player," NHL referee supervisor Don Koharski said.
Quick stopped all eight first-period shots he faced, including an in-close chance by Vincent Lecavalier. He was pulled in the second period of the loss at Atlanta.
"I think it's a huge lift for him, personally," Los Angeles captain Dustin Brown said.
Niittymaki made a nifty glove save on a shot by NHL points leader Kopitar from near the crease just over a minute into the second. Kopitar assisted on Doughty's goal for his 17th assist and 31st point.
"I think everybody was just competing a lot harder," Kopitar said. "We were winning loose puck battles and everybody played for each other."
Both teams had 18 shots through two scoreless periods.
"The biggest thing is just moving forward and building off this win here," Quick said.
Slovakia national team GM Peter Bondra and coach Jan Filc were at the game. Kings C Michal Handzus is a candidate for the country's 2010 Olympic team. Meszaros was on the 2006 squad and could play again in February. ... Los Angeles D Rob Scuderi left in the first because of a lower body injury. ... Malone has scored a goal in three straight games.
Copyright by STATS LLC and The Associated Press
TAMPA, FL - NOVEMBER 14: Ryan Malone #12 of the Tampa Bay Lightning walks to the ice for pre-game skate before...