Elliott stopped 11-of-12 shots in the first period as the Blues were grabbing the lead on Tarasenko's eighth goal of the season, which came only 9 seconds after Brenden Dillon had forged a 1-1 tie for the Stars.
Kari Lehtonen started in net for Dallas but was lifted in the third period after yielding five goals on just 18 shots.
Dan Ellis stopped 6-of-7 shots the rest of the way.
Scarlett Johansson steps out with fiance after pregnancy reveal
Interpol investigating stolen passports on missing flight