The Blues surrendered three one-goal leads during the contest, including a 3-2 advantage in the third period when Winnipeg's Dustin Byfuglien scored on a power play to knot the score with 6:17 remaining.
After a scoreless overtime, the contest moved to shootout.
Oshie beat Al Montoya onto a wrist shot in the second round, and after Brian Elliott stopped the Jets' Andrew Ladd, Tarasenko fired a snap shot in the third that found the far right corner of the net.
Elliott finished with 28 saves as the Blues entered the Winter Olympics break with a third win in four games.
Mark Scheifele scored twice and Blake Wheeler provided a pair of assists for Winnipeg, losers of two straight.
Montoya turned aside 23-of-26 shots.