Brian Elliott stopped all three attempts by the Coyotes in the shootout, bringing St. Louis two crucial points in the playoff race.
The Blues began the night in seventh place in the Western Conference, one point in front of Columbus and one behind Minnesota.
David Perron scored for St. Louis midway through the first period and with time beginning to run out it appeared that was going to be the only goal the Blues needed.
Elliott's bid for a shutout, however, was spoiled with 1:07 left in regulation when Radin Vrbata scored on the power play.
Elliott made 33 saves while Chad Johnson stopped 29 shots for the Coyotes. The Phoenix post-season hopes have faded with three straight losses.