It was Dickerson's second RBI triple of the game and came after the Rockies handed St. Louis closer Edward Mujica his fourth blown save of the season.
St. Louis saw its lead over Pittsburgh in the National League Central reduced to one game with nine to play.
With both bull pens extended due to the length of the game, Fernando Salas (0-3) came out in the 15th for his second inning of work.
He gave up a one-out single to LeMahieu and Dickerson followed with his triple to right.
That made a winner out of Rob Scahill (1-0), who picked up his first major league victory by working his way out of a bases-loaded jam in the top of the 15th. Pete Koza grounded into a double play to end the St. Louis threat.