After Stewart gave the Blues a 1-0 lead on a long rebound to the bottom of the right circle, a series of defensive miscues led to four of the next five goals.
Nineteen seconds into the second, the Blues failed to clear their zone after goalie Jake Allen settled the puck behind the net. Martin Hanzal stole the puck from Barret Jackman and pushed it to Doan in the right faceoff circle to make it 2-1.
Berglund tied the game after Mike Smith blocked a shot by Perron high in the air. The puck landed just outside the right faceoff circle, where Berglund pounced after two Coyotes skated past, and beat Smith to his glove side.
He added his second by kicking a pass from Perron from his skate to his stick and backhanding a shot that deflected off Derek Morris' stick into the top of the net.
Sobotka got a pass from Stewart and fired into the open right side of the net to make it 5-2.
Pietrangelo scored with 4:56 left. Yandle capped the scoring with his goal with 25 seconds to go.
NOTES: Smith has allowed four or more goals in five of his past six games. ... Moss' goal was the Coyotes' first short-handed goal of the season. ... Stewart was tied for the Blues' goal-scoring lead for just under 11 minutes, until Berglund scored his 10th and 11th. ... St. Louis had been outscored 17-0 in the second period since Feb. 1 until its three-goal outburst. ... The Blues have won three straight road games against the Coyotes for the first time since 1999-2000. ... Coyotes LW Radim Vrbata missed his ninth straight game with a lower-body injury.
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