Johnson took a pass from Alexander Steen right in front of the goal at 13:13 of the final period and knocked it past Jonathan Quick before crashing into the net himself.
The goal, Johnson's fifth of the season, gave the Blues their second win in three tries.
Patrik Berglund scored for St. Louis on a power play at 11:56 of the first period to open the scoring. Jaroslav Halak made 26 saves on 27 shots.
Quick, who had missed the previous three games, made 24 saves. Kyle Clifford scored a second-period goal for LA that tied the score 1-1.