Mike Richards and T.J. Oshie both scored in the second round, and Dustin Brown and Jaden Schwartz both came up short in the third, before Carter beat Brian Elliot. Chris Stewart was unable to keep the shootout going, hitting a backhander just wide of the left post.
Jonathan Quick finished with 35 saves for his eighth shutout of the season.
The Kings moved up to third place in the Western Conference and took over eighth-place from Phoenix in the Pacific Division, while the Blues -- who lead the Central Division and the Western Conference -- lost for the fourth time in five games.
Elliot stopped 37 shots and posted his seventh shutout of the season.