T.J. Oshie had a goal and an assist for the Blues (36-15-4), who remained four points behind the Nashville Predators in the race for first place in the Central Division. The Predators defeated the Winnipeg Jets 3-1 and lead the NHL with 80 points.
Tampa Bay (34-17-6) lost at home for the second time in the past three games. The Lightning and Montreal Canadiens each have 74 points, the most in the Eastern Conference, but the Lightning have played two fewer games.
After the morning skate, several Blues players mentioned the need to play better than they did in a 2-1 overtime win against the Lightning in St. Louis on Feb. 3. Coach Ken Hitchcock said his team "stole" two points that night despite being outplayed.
It was a much different story Thursday, when the Blues came with pressure all night and smothered the Lightning's defense in a dominating performance that gave Hitchcock his 693rd coaching win, moving him past Dick Irvin Sr. for fourth on the all-time list.
"You can't get a lot of wins in this league without good players," Hitchcock said. "I've had good players and good coaches. In all the places I've been there have been just two coaches that have changed. The coaching staffs have stayed the same and familiar with each other. I think that's the thing I'm proudest of; we've stayed together."
St. Louis established the pace early and jumped out to a 2-0 in the first 4:47 of the first period, then put the game away with scored three goals in a three-minute span in the second period. The Blues outshot the Lightning 44-29.
"Guys were making plays all over the place," forward David Backes said. "It was a great effort by our guys to come back and pour in a lot tonight, especially after a game in our building where they really took it to us the whole game."
Steen set up Oshie's goal at 1:50 in the first period after he escaped the forecheck of Lightning defenseman Anton Stralman to create a 2-on-1. Steen sent a perfect centering pass to the streaking Oshie, who sent a wrist shot high above the stick of goalie Ben Bishop.
"I thought we did a good job of jumping on them early," Steen said. "The big thing is we're skating and moving to get open and that's what's generating the seams to make these passes and generating opportunities. Jori [Lehtera] had a few passes to me that I know I should have finished on."
Less than three minutes later Steen was at it again, this time on the Blues' power play. He sent a pass between two Lightning defensemen right to the stick of forward Dmitrij Jaskin, who easily scored on a breakaway to make it 2-0 at 4:47.
"I thought the first period was the best we've played on the road for a long time," Hitchcock said. "We've had two complete games on the road now. We managed the puck well and got a lot of contributions. They were quicker in every aspect of the game the first time we played them but we made some adjustments and credit to the guys for executing it."
Tampa Bay's Tyler Johnson made it 2-1 with his 20th goal of the season at 8:03 of the second period. He fired a one-timer past St. Louis goalie Brian Elliott off a pass from Nikita Kucherov for a power-play goal.
But Steen made it 3-1 at 12;15 when he capitalized on a turnover by Lightning center Steven Stamkos and beat Bishop from close range for his 19th of the season.
Backes pushed the lead to 4-1 at 14:11 with 19th goal of the season. Oshie made a pass between his legs from behind the goal that found Backes in the slot, where he sent a high wrist shot past Bishop's glove.
"I just saw it come to me in the slot and I didn't want to screw it up," Backes said. "Luckily it ended up in the back of the net. It was a pretty impressive play.
Paul Stastny scored 72 seconds later to make it 5-1, and after goals by Brett Connolly in the second period and Stamkos early in the third got Tampa Bay within two, Vladimir Tarasenko hit the empty net with 1:05 remaining. It was his 29th goal of the season.
"We didn't play well defensively," Stamkos said. "Other than the third period we didn't play well at all. We gave pucks away, we didn't we battles….we did the opposite of the last time we played these guys. This was unacceptable, especially in front of our fans. We have to learn a quick lesson from this that we can't just show up for games and win."
Elliott finished with 26 saves. Bishop made 29 saves through two periods, and Andrej Vasilevskiy stopped all nine shots he faced in the third period. Lighting coach Jon Cooper said he pulled Bishop after "facing a game's worth of shots." Bishop was also pulled on Saturday after giving up three first-period goals in a 4-2 loss to the Los Angeles Kings.
"Our start was horrific and they came ready to play," Cooper said. "They clearly knew what happened to them the last time we played them, even though they found a way to win that game. They came in here [angry] and they played like it."