Berglund gave the Blues a 3-2 lead at 7:59 of the second period with a nifty play, kicking a pass from David Perron from his skate to his stick and backhanding a shot that deflected off Derek Morris' stick into the top of the net.
"I almost blacked out. I don't know what happened," Berglund said. "It was a really quick pick up and I turned around to try and protect it. I was getting closer and closer to the net so I just turned around and threw it backhand. I looked up and didn't really know where I was aiming."
Berglund had tied the game at 5:47 of the period 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.
"There were times, if you look at the game, where they're scoring goals with red jerseys around," Phoenix Coyotes coach Dave Tippett said. "We're just not doing much. That's concerning."
"Points are at a premium right now, so we'll take the two points," Stewart said.
David Moss scored a short-handed goal, Keith Yandle had a power-play goal and Shane Doan added his second goal in three games for the Coyotes, who have lost two straight.
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.
Boyd Gordon stripped the puck from Pietrangelo just inside the Coyotes' blue line and sent Moss ahead with a quick pass for a breakaway goal late in the opening period to tie the game 1-1.
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.
Jaden Schwartz took the puck off Kyle Chipchura's stick, skated in front and slid a quick pass to Nichol for a one-timer and a 4-2 Blues lead midway in the second period.
Sobotka got a pass from Stewart and fired into the open right side of the net to make it 5-2.
"Some of the turnovers are really concerning because it's the same people doing it in the same place and the same part," Tippett said. "We'll continue to address it, but if it doesn't get cleaned up, we'll have to try different options."
Pietrangelo scored with 4:56 left. Yandle capped the scoring with his goal with 25 seconds to go.
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.
Copyright by STATS LLC and The Associated Press
Arena: Gila River Arena Location: Glendale, Arizona
Referees: Dan O'Halloran, Kyle Rehman Linesmen: Shane Heyer, Thor Nelson
Attendance: 11,482 (67.0% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Chris Stewart (9)
Assists: Alex Pietrangelo, David Perron
David Moss (3)
(Shorthanded) Assist: Boyd Gordon
Shane Doan (7)
Assist: Martin Hanzal
Patrik Berglund (10)
Assists: Barret Jackman, Chris Stewart
Patrik Berglund (11)
Assists: David Perron, Chris Stewart
Scott Nichol (1)
Assist: Jaden Schwartz
Vladimir Sobotka (4)
Assists: Chris Stewart, David Perron
Alex Pietrangelo (5)
Assists: T.J. Oshie, David Backes
Keith Yandle (5)
(Power Play) Assists: Steve Sullivan, Derek Morris