"We just dig in," said Blues defenseman Eric Brewer, who gave his team a 4-2 lead in the second period with his fourth goal of the season. "Everyone is committed to this work ethic."
St. Louis, which had lost seven of nine overall, is 10-2-2 at home. The Kings lost for the second time in six games.
Ty Conklin made 25 saves and improved to 6-1 in his career against Los Angeles.
Steen took a drop pass from Jay McClement and whistled a shot past Jonathan Quick for his fourth game-winning goal this season. Quick stopped a career-high 51 shots in his previous outing, a 5-0 win over Detroit on Monday.
"A good shot beats a good goaltender," St. Louis coach Davis Payne said.
The Kings outshot St. Louis 12-7 in the third period and had several chances to break the tie. Conklin made two big saves in the middle of the period including a stop of Anze Kopitar, who scored Los Angeles' second goal.
"We had some good things happen in that third period," Kings coach Terry Murray said. "We had four or five good shifts but just didn't finish."
Kopitar said a defensive mistake led to Steen's winning goal.
"You've got to play consistent for 60 minutes," Kopitar said. "You can't have defensive breakdowns like that in the third period."
Kopitar scored his 14th of the season to give the Kings a 2-1 edge early in the second. Kopitar, who has six goals and four assists in the past six games, converted on a 5-on-3 advantage.
Ryan Smyth put Los Angeles ahead in the first period, but McClement answered for the Blues.
Kings goalie Jonathan Quick made his eighth successive start. ... LW Marco Sturm, acquired from Boston for future considerations, practiced with the Kings for the first time Wednesday. He will likely make his Los Angles debut next week. ... Los Angeles defenseman Willie Mitchell sustained a lower-body injury early in the second period and did not return. ... The Blues killed off 16 consecutive penalties before Smyth's power-play goal in the first period. ... St. Louis had its 14th successive sellout this season.