Niskanen took a pass from Kris Letang and scored on a one-timer from the left side at 7:49.
Jeffrey then scored his first goal since Feb. 18, 2012, when he redirected a shot from Matt Cooke at 15:15 to give the Penguins a 3-1 lead.
Malkin remained down on the ice for several moments, but skated off the ice on his own.
The Panthers beat the Philadelphia Flyers on Thursday for their first win in five games, and Pittsburgh got off to a strong start on Friday.
The Penguins outshot Florida 12-3 in the first period and had two power plays to the Panthers' one, but the game was still scoreless.
The three shots by the Panthers were the fewest the Penguins have allowed in a period this season.
Weaver scored his first goal since Jan. 25, 2011, from the point to give Florida a 1-0 lead at 8:25 of the second period.
Fifty-eight seconds later, the Penguins got their third power play of the game and Kunitz chipped the puck from under the glove of Markstrom to tie the game at 1-1 at 10:41 of the second.