Mathieu Garon stopped 35 shots to help the Kings improve to 8-0-4 in their last 12 home openers.
Lubomir Visnovsky, Michael Cammalleri and Brent Sopel also scored for the Kings, whose last loss in a home opener was a 5-2 defeat to Vancouver on October 6, 1993.
Kopitar, a 19-year-old centre from Slovenia, had a pair of goals in his first NHL game - a 4-3 loss at Anaheim on Friday night.
The loss was the Blues' franchise-record 11th straight on the road. St. Louis lost nine straight road games to end last season and opened with a 5-4 overtime setback at San Jose on Thursday night.
Taking a pass from Rob Blake along the left side wall during a Kings second-period power play, Kopitar - Los Angeles' top pick, 11th overall, in the 2005 draft - swiftly moved the puck to Frolov in the corner. Visnovsky stepped into a centring pass from Frolov and one-time a slap shot past Blues goalie Manny Legace for his second goal of the season and a 1-0 lead for the Kings.
St. Louis tied the game 79 seconds later as Martin Rucinsky beat Garon with a 50-foot slap shot from the left point on the power play.
Los Angeles responded 3:04 later. As Kopitar crashed the net and toppled over Legace, Frolov swiped in a lose rebound, putting the Kings ahead for good.
Cammalleri provided an insurance goal 13 seconds into the third period, and Kopitar assisted on Sopel's goal with 4:18 to play.
Notes: Blues D Barret Jackman left the game with a sore left knee and did not return. ... D Oleg Tverdovsky made his debut for the Kings after he was a healthy scratch in Friday's game.