St. Louis improved to 7-1 since Mike Yeo took over as coach, which is the best start for a coach in Blues history.
Tarasenko gave the Blues a 3-2 lead with a power-play goal 58 seconds into the third period. It was the Blues first goal with the man-advantage in four games.
Steen's power-play goal five minutes into the period made it 4-2 and stood as the game-winner. It came on the second man-advantage of a double-minor given to Sedin for high-sticking.
Less than two minutes later, Sutter deflected Alexander Edler's shot past Allen to cut the deficit to one.
Paajarvi backhanded Shattenkirk's shot off the end boards to give the Blues a 1-0 lead at 4:30 of the first period. It is the sixth straight game in which the Blues have scored first.
Lehtera scored his first goal since Jan. 14 to give the Blues a 2-1 lead late in the first. The Canucks unsuccessfully challenged the goal, claiming the Blues were offside.
Sedin scored off a nifty pass from his brother Daniel Sedin to tie the game at 8:27 of the second period. The goal was scored 19 seconds after Shattenkirk's shot trickled in past Markstrom that appeared to give the Blues a two-goal lead, but the referees had blown it dead believing that Markstrom had stopped it.
NOTES: The Blues improved to 7-2 in the second game of back-to-back sets. ... With their win Wednesday at Detroit, the Blues swept a five-game road trip for the first time in franchise history. ... Horvat (bone bruise) and C Brandon Sutter (wrist) returned to the Canucks' lineup after each missing a game.
Canucks: host Calgary on Saturday.
Blues: travel to Buffalo on Saturday.