Moore was playing his first game after missing the previous 11 with an upper-body injury.
Alex Laferriere had a goal and an assist for the Kings (21-16-13), who have won two straight after four consecutive losses (0-1-3). Darcy Kuemper made 24 saves.
Adrian Kempe also scored in the shootout for the Kings. Dalibor Dvorsky scored for the Blues in the third round to keep the shootout alive.
Jordan Kyrou scored twice for the Blues (19-24-9), who have lost four straight (0-3-1), including 3-2 at the Dallas Stars on Friday. The loss also ended St. Louis' four-game home winning streak. Joel Hofer made 23 saves.
Moore put in a rebound at 11:34 of the third period to give the Kings a 4-3 lead with his 100th NHL goal. The Blues challenged for offside, but the goal was upheld after video review.
Kyrou tied it 4-4 at 17:50 for the Blues when he one-timed the rebound off Justin Faulk's point shot with Hofer pulled for an extra attacker.
Taylor Ward gave Los Angeles a 1-0 lead at 4:31 of the first period when Alexey Toropchenko’s outlet pass along the left boards was deflected and intercepted at the blue line, and Ward’s shot from between the hash marks in the slot beat Hofer to the stick side.
Brian Dumoulin’s first of the season made it 2-0 at 1:11 of the second period on a one-time slap shot from the left point through a screen.
The Blues would rally for three goals in 5:56 to take a 3-2 lead.
Dvorsky put St. Louis on the board at 2:14 to make it 2-1, then Brayden Schenn’s power-play goal at 6:30 tied it 2-2, and Kyrou’s wraparound goal at 8:10 gave the Blues their first lead at 3-2.
Laferriere tied it 3-3 at 10:18 on a short side shot just inside the post.
