The Lightning (13-11-1), who lost their fourth in a row and fifth in six games, got two goals from Cedric Paquette. Tyler Johnson and Nikita Kucherov also scored. Andrei Vasilevskiy made 12 saves before being pulled in the second after allowing four goals. Ben Bishop replaced him and made nine saves.
"I thought we deserved points," Lightning coach Jon Cooper said. "I'm not going to sit here and say that we deserved to win the game, but we definitely deserved points out of this game."
Tarasenko gave the Blues the lead 1:39 into the game with his 11th goal, then made it 2-0 on the first of three power-play goals on a one-time shot from the right circle at 9:37.
Paquette got the Lightning within 2-1 at 10:39, but Shattenkirk made it 3-1 with a wrist shot off Paquette's skate at 13:17.
Shattenkirk scored the Blues' third power-play goal on a slap shot from the high slot at 1:44 of the second period that chased Vasilevskiy.
Johnson scored for Tampa Bay at 14:25 to make it 4-2, and Kucherov's power-play goal 1:41 into the third cut the Blues' lead to 4-3.
Tarasenko's third goal and fourth point came at 8:11 against Bishop to give the Blues a 5-3 lead, but Paquette's second of the game at 11:56 made it 5-4.
Tampa Bay couldn't score the equalizer despite outshooting St. Louis 12-6 in the third.