Kessel gave Toronto a 2-0 lead in the first period with his NHL-leading 12th goal of the season, but the Blues fought back to tie it with 1:56 left in regulation.
Kessel then beat Jaroslav Halak in the first round of the shootout and Ben Scrivens stopped all three St. Louis efforts to hand the Blues their first overtime loss of the season.
Buffalo, Boston, Washington and Dallas are the only teams left without a setback in overtime.
The Maple Leafs, who had lost their previous two outings, became the first team in the Eastern Conference to reach 10 victories.
Scrivens, in the third start of his rookie season, made 37 saves. Halak stopped 20 shots.