Kucherov is third in the NHL with 22 goals and leads the League with 54 points. He has 10 points (five goals, five assists) in his past five games.
Nicholas Paul had a goal and two assists, Steven Stamkos had three assists, and Andrei Vasilevskiy made 28 saves for the Lightning (15-13-5), who have won two of three.
"We have to simplify our game, it sounds like a cliché," Tampa Bay coach Jon Cooper said. "When you do those things, you're hoping to put teams under duress, making the game a little more unpredictable in their defensive zone. If you're around it and shooting enough, good things usually happen."
Justin Faulk scored, and Jordan Binnington allowed five goals on 23 shots before being pulled in the second period for the Blues (15-15-1), who lost for the first time under coach Drew Bannister (2-1-0). Joel Hofer made nine saves in relief.
"It was a poor start, and we weren't able to get any kind of forecheck," Bannister said. "I think we had 10-13 turnovers in the [defensive] zone and 10 turnovers in the offensive zone.
"Biggest thing that stuck out to me is that it was too easy to get pucks to our net. We weren't willing to get into shooting lanes. We didn't get sticks into shooting lanes."