Only 45 seconds into the extra frame, St. Louis came in on a 3-and-1 break and Stewart got control of the puck in the slot. He faked Ducks goalie Jonas Hiller to the ice and then backhanded it into the open net.
It was the fifth goal in five games and a team-leading 14th of the season for Stewart, who boosted the Blues to their third straight win.
Kris Russell also scored and rookie goalie Jake Allen improved to 8-1-0 on the season with 27 saves.
The Ducks, meanwhile, saw their five-game winning streak snapped.
Hiller finished with 19 saves and Saku Koivu scored Anaheim's only goal.