Stamkos reached 594 career goals, seven behind Jari Kurri for 21st in NHL history. His four goals tied a Predators record (Filip Forsberg, Rocco Grimaldi, Eric Nystrom) for most in a game.
Luke Evangelista had three assists for the Predators (12-14-4), who have won six of their past eight games. Juuse Saros made 24 saves.
Philip Broberg had two assists for the Blues (11-14-7), who had won two of their previous three. Jordan Binnington was pulled after allowing six goals on 25 shots through two periods. Joel Hofer made six saves in relief.
Stamkos gave the Predators a 1-0 lead at 8:22 of the first period. He knocked the puck out of mid-air on a rebound of his own shot on a 2-on-1 rush with Matthew Wood.
Stamkos made it 2-0 at 11:19 on a rebound in the slot on a shot from the point by Roman Josi.
Hugh McGing scored his first NHL goal for the Blues to make it 2-1 at 4:17 of the second period on a shot from the right face-off circle on a pass from Robby Fabbri.
Ryan O’Reilly gave the Predators a 3-1 lead at 8:38 on a one-timer.
Forsberg extended the Predators lead to 4-1 at 11:43 on a 2-on-1 rush with Evangelista.
Stamkos completed his 15th NHL hat trick to make it 5-1 at 12:06 on a passing attempt that deflected in off of Blues defenseman Justin Faulk.
Robert Thomas scored for the Blues to make it 5-2 at 13:56 on a wrist shot from the right circle that beat Saros on the blocker side.
Stamkos gave the Predators a 6-2 lead at 15:33 on a rebound of a shot from Josi that was redirected by Wood.
Bunting made it 7-2 at 7:36 of the third period on the power play.