Bo Horvat, J.T. Miller and Micheal Ferland scored for the Canucks, who won for the first time on the road in three tries this season.
Vladimir Tarasenko, Alex Pietrangelo and Robby Fabbri scored for the defending Stanley Cup champions, who have dropped three straight.
Leivo lifted a shot over Jordan Binnington in the bottom half of the sixth shootout round after the first 11 shooters came up empty.
Hovath jammed the puck in off a scramble in front of the net to tie the game at 3-3 with 7:55 left in the third period.
Tarasenko and Pietrangelo scored 47 seconds apart in the second period to give the Blues a 3-1 lead.
Pietrangelo ripped a one-timer from the top of the faceoff circle off a pass from Oskar Sundqvist at 2:47. Tarasenko then jumped on the rebound of a shot from Jaden Schwartz for his second goal in as many games.
Ferland tied the game 1-1 on the rebound of shot by Brandon Sutter. Pietrangelo answered 42 seconds later
St. Louis has points in six of seven games this season.
NOTES: The Blues visited the White House on Tuesday. Since 2007, teams are 7-2-2 in their first game after visiting the White House. ... Vancouver G Jakob Markstrom is not with the team because of a family emergency. He has missed the last two games. ... St. Louis C Brayden Schenn had a five-game goal streak snapped. ... Blues RW Robert Thomas returned to the lineup after missing the past five games with an upper-body injury that was not related to his left wrist injury that required offseason surgery. ... St. Louis Cardinals pitcher Adam Wainwright and catcher Yadier Molina were in attendance, two days after they were eliminated from the playoffs by the Washington Nationals.
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