St. Louis led 2-1 entering the third period before the Stars struck three times within 8 minutes. Dallas' third-period goals were against Niklas Lundstrom. Benn, who had only three goals last season, broke a 2-2 tie on the power play.
The Blues' Patrick Berglund assisted on Ryan Whitney's power-play goal and beat Dallas goalie Anders Lindback on a breakaway. Ian Cole scored in the third period for St. Louis.
Each starting goalie, the Blues' Brian Elliott and the Stars' Kari Lehtonen, allowed one goal and made 11 saves.
Dallas led 1-0 after one period on a goal by 34-year-old Vernon Fiddler.
Dickinson, 19, was Dallas' first-round draft pick in 2013.