St. Louis was headed toward its fourth consecutive victory before Rick Nash tied it with 14 seconds left in regulation. It was Nash's team-leading 16th goal.
Stempniak, who scored his 18th goal earlier in the game, went high over Fredrik Norrena and Weight made a move just in front of the net before sliding the puck in.
Neither of the Blue Jackets' shots in the shootout had a chance. Legace tipped Nikolai Zherdev's shot high and wide and David Vyborny fired his directly into the Blues goalie.