Malkin added an assist to bring his NHL-leading points total to 77 and Staal also added a helper as the Penguins rolled to their fifth win in seven games.
Chris Kunitz, Matt Cooke and Pascal Dupuis also found the net while Steve Sullivan and James Neal each added three assists for Pittsburgh, which scored three times within a 4 1/2-minute span in the first period to grab control.
Mark-Andre Fleury turned aside 34-of-35 shots to earn the victory.
Teddy Purcell scored for the Lightning.
Tampa Bay's Dwayne Roloson yielded all eight of Pittsburgh's goals on 34 shots.
Notable deaths of 2014 [PHOTOS]
Michael Sam cut from the Dallas Cowboys