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.
His early game outburst set the tone for the rest of the contest.
"We shot the puck at the right times, people were in position," St. Louis coach Ken Hitchcock said. "It was great to see."
Tarasenko, known for his speed and puck-handling skills, scored his second goal on a rare one-timer off a perfect pass from Alexander Steen, who added three assists in his return after missing six games with a shoulder injury.
The soft-spoken, Russian-born Tarasenko has nine goals and eight assists in his last 13 games.
"It's good to see him playing like this," Steen said. "He's been lethal for us. His hat trick goal was big, a timely goal."
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.
Nikita Kucherov and Tyler Johnson also scored for Tampa Bay. Paquette's second goal trimmed the deficit to 5-4 with 8:02 remaining in the third period.
"I thought we deserved points," Tampa Bay coach Jon Cooper said. "I'm not going to sit here and say we deserved to win the game. But we definitely deserved points."
St. Louis has won its last five at home against the Lightning, who last won in St. Louis on Dec. 18, 2009.
The Blues are 9-0-2 in their last 11 home games.
Tampa Bay goalie Andrei Vasilevskiy gave up four goals on 16 shots and was replaced by St. Louis native Ben Bishop early in the second period.
"I thought we gave it an honest effort," Paquette said. "It was a pretty good game. But (to lose), it's disappointing."
NOTES: The Lightning recalled RW Joel Vermin and LW Michael Bournival from Syracuse of the AHL. ... St. Louis leads the overall series with Tampa Bay 21-11-3. ... Tampa Bay D Anton Stralman returned to the lineup after missing nine games due to an upper body injury. ... Blues LW David Perron had a career-best eight-game point streak snapped. ... Bishop did not start in back-to-back games for the first time this season. ... St. Louis C Paul Stastny played in his 700th NHL game.
Lightning: Return home after three-game trip to host Washington on Saturday.
Blues: Host Winnipeg on Saturday in the fourth of five straight games at home.