The Blues have won three in a row and six straight at home.
Tampa Bay lost its season-high fourth consecutive game.
Jake Allen made 22 saves in improving to 12-3-3.
Tarasenko, who leads the team with 13 goals, recorded his fourth career hat trick. He scored twice in the game's opening 9:37 to stake his team to a 2-0 lead.
Shattenkirk, who also had two assists, added two power-play goals, the second coming just 1:44 into second period for a 4-1 lead.
Cedric Paquette scored twice for the Lightning, who have been outscored 19-9 during the four-game skid.
