Stamkos, twice the winner of the Maurice Richard Trophy as the NHL's leading goal scorer, tallied once in each period and had a goal both shorthanded and on the power play.
Valtteri Flippula added two goals for the Lightning, who improved to 3-1 by equaling their entire offensive output from their first three games.
Florida has one win in four tries.
Stamkos gave Tampa Bay a 2-0 lead in the first period by tipping in a shot from Martin St. Louis with the Lightning playing a man down. He also tallied midway through the second period and completed the hat trick on the power play with 7:35 remaining in the third.
Ben Bishop had to make just 17 saves to pick up the easy win. Jakob Markstrom gave up the first six Tampa Bay goals on 24 shots before being replaced by Scott Clemmensen.
The first shot Clemmensen saw came from Stamkos and wound up in the net.
Celebrity Families of 2014 [PHOTOS]