Stempniak collected the puck in the neutral zone and fired a low wrist shot from the left circle that got through the pads of Jean-Sebastien Giguere.
It was the second straight overtime victory for the Blues, who have won two out of their last three games. David Backes and Steve Regier also scored for St. Louis.
The win went to Manny Legace, who made 26 saves.
Anaheim, loser of five out of six, got goals from Chris Kunitz and Scott Niedermayer. Giguere stopped 25 shots.
St. Louis, in last place in the Central Division, has not won a game in regulation time since Oct. 25.