"We want to win every game, but, definitely, when you got a case of the skids you want to kick it in the butt as soon as you can," said Kuemper, who got his first shutout with the Kings and eighth in the NHL. "It's obviously a huge step for us, and now we just [have] to get back to playing our game."
James Reimer made 16 saves for the Panthers (7-10-2), who had a two-game winning streak end. Florida has yet to win three straight.
Brodzinski scored at 2:23 of the third period, his first goal in his 11th NHL game. He is the fifth Kings rookie to score his first NHL goal this season, joining forwards Alex Iafallo and Michael Amadio and defensemen Oscar Fantenberg and Kurtis MacDermid.
"It just tells you the amount of young people we've had in our lineup, which is kind of exciting to see," Kings coach John Stevens said.
Brodzinski, 24, was recalled from Ontario of the American Hockey League on Thursday. He was with Los Angeles for its first five games before being sent to the AHL.
"He went back and really seemed to get his game in order," Stevens said. "He is really starting to play with the pace that he can play with, and what we've seen now since he's come back he's a lot more confident young man. You do the work, you do the preparation, and confidence seems to come with it. That was a big-time play he made tonight, but it's kind of what we thought Jonny was capable of so, it's good to see."
Brodzinski's former high school linemate in Blaine, Minnesota, Panthers forward Nick Bjugstad, was on the ice when he scored.
"He's got a few more than me this year so I've got to catch up," Brodzinski said of Bjugstad, who has four goals this season and 67 in 299 NHL games.
Toffoli gave Los Angeles a 1-0 lead at 8:14 of the first period. He got onto the ice as an extra attacker on a delayed penalty and scored off a pass from Drew Doughty.
Toffoli had gone three games without a point after getting eight (six goals, two assists) in the previous seven.
Kuemper made 14 saves in the first period, including a sequence when he denied Aleksander Barkov and Evgenii Dadonov in succession.