Tarasenko flipped the puck over the shoulder of Jack Campbell and into the upper left corner of the net.
First-round draft pick Valeri Nichushkin scored two goals to give Dallas leads of 4-3 and 5-4.
The Blues tied it 5-5 on Ryan Reaves' goal with 8:57 remaining in the third period.
Dallas took a 2-0 lead 4 minutes into the game on goals by Erik Cole and Kevin Connauton, but the first period ended in a 2-2 tie after newly acquired Maxim Lapierre scored twice for St. Louis.
Tyler Seguin scored on a second-period power play in his Stars debut.