The Maple Leafs (8-7-0), who went 1-3-0 on a four-game road trip, have lost six of eight.
Tyler Bozak scored twice, Andreas Borgman had a goal and an assist, Connor Brown scored, and Matt Martin had three assists for the Maple Leafs. Frederik Andersen made 26 saves.
"We had a tough trip and played good teams and all that, but obviously we're not in a situation that we're handling this very well," Toronto coach Mike Babcock said. "So we're not playing good enough, and like I said, get ourselves regrouped. We've got to be [ready] against a team that seems to win every night in their building here."
Borgman, who took Martin's feed inside the blue line, skated into the right circle and beat Allen with a wrist shot high stick side at 9:16 of the first period to give Toronto a 1-0 lead.
Tarasenko tied it 1-1 on a rebound goal at 2:33 of the second period. Edmundson put the Blues ahead 2-1 with another rebound goal, this time from the left of Andersen, at 7:10.