Steen's goal early in the third period gave the Blues a seemingly commanding four-goal lead, but the Phoenix Coyotes rallied to make it close.
In the end, the Blues held on 4-3 on Friday night.
"They didn't go away," Steen said. "They battled back, worked hard, but we got the 'W.'"
Steen had a goal and an assist, Jaroslav Halak stopped 30 shots and the Blues won their fifth straight game. Brad Boyes, Erik Johnson and Alex Pietrangelo added goals and Barret Jackman had two assists to stake the Blues to a 4-0 lead early in the third period.
Shane Doan, who picked up his 700th career point, had two goals for Phoenix, the first at 6:20 of the third period to end Halak's shutout streak at 129 minutes, 29 seconds. Halak also has shutout streaks this season of 160:08 and 153:03.
The Blues are playing much like they did early in the season when they got off to a franchise-best 9-1-2 start. Getting injured players back in the lineup certainly helps, but the play of Halak and backup Ty Conklin has fueled the latest stretch of wins.
"We wanted to finish the year on a winning note," Halak said. "Fortunately for us, we got a 4-0 lead. We got a little comfortable in there, giving them opportunities to score goals, but they all count. We got the two points and that's all that matters right now."
Lauri Korpikoski also scored for the Coyotes, have been outscored 14-4 in the previous four road games, going 1-3-0 in that stretch.
Despite being outshot 14-6 in the first period, the Blues grabbed the lead when Boyes converted a cross-ice feed from Steen and beat Ilya Bryzgalov with a shot through the goalie's leg pads with 5:52 left.
"We created some chances," Coyotes coach Dave Tippett said. "The one chance they got, they capitalized on. We had a couple of chances very similar to that and didn't capitalize on. If we get a lead in that first period, it could be a different outcome. We feel like we played better"
Johnson's short-handed goal 7:23 into the second period put the Blues up 2-0. Backes drove hard to the net, got a shot off that Johnson collected in the right circle and snapped past Bryzgalov.
"That was probably the moment that allowed the game to tilt a little bit more in our favor," Blues coach Davis Payne said. "Great individual play, great read, and now all of the sudden, we feel like we're in more command of the game."
Pietrangelo's shot from the top of the blue line on the power play got through Bryzgalov as Backes was setting a screen in the slot to give the Blues a 3-0 lead with 1:31 left in the second.
Early in the third, Steen picked off an errant pass in the Blues' zone and skated in on a 2-on-1 with Eric Brewer and beat Bryzgalov with a low shot to the stick side.
Doan put the Coyotes on the board by crashing the net and backhanding a shot that squirted through Halak's pads and break the shutout. Phoenix made it interesting, with Korpikoski scoring from the slot with 1:27 remaining and Doan tipping in a goal with 20.9 seconds left.
"We had control of the play and you give Halak credit," Doan said. "He was really good tonight. When he's the best player on the ice, it's going to be tough."
Halak faced the Coyotes for the first time in his career. ... Steen has at least a point in 12 of 15 games. ... The Blues winning streak is two off a season-best seven-game winning streak from Oct. 22-Nov. 7. ... Coyotes C Martin Hanzal (lower-body), LW Taylor Pyatt (upper-body) and RW Radim Vrbata (lower-body) returned to the lineup. ... Since returning from injury Nov. 21, Doan has 20 points in 18 games. He also has 44 points in 59 career games against St. Louis.