The Blues had lost their previous two games and the defending Western Conference champion Canucks have dropped two in a row.
Alex Burrows put Vancouver in front with 6:47 left in the first period, but T.J. Oshie produced the game's next two goals to give St. Louis the lead.
Oshie scored on a backhander with 5:10 remaining in the opening period and added his third of the season 1:28 into the second.
St. Louis kept the lead until Ryan Kesler scored on the power play 23 seconds into the final period. The game-winner by Backes was his fourth goal of the season.