Shattenkirk gave the Blues their first lead of the game with 7:14 left when he took a pass at the point, skated into the left faceoff circle and took advantage of a screen from T.J. Oshie to score his first goal since Jan. 21.
Berglund capped the scoring with an unassisted goal with 1:39 left, jumping on a loose puck after Yandle was apparently tripped between the circles in the St. Louis zone and skating the length of the ice before scoring on a sharp wrist shot.
"I think that in the last few games we've had a lot of chances, but we haven't been able to capitalize," Shattenkirk said. "We did a great job of going down 2-nothing and staying with our game plan."
Jeff Halpern scored his 150th career goal and Bissonnette added his first goal since Dec. 14 for the Coyotes, who have lost four straight and five of six.
"I still look and there are good things that we are doing," said Coyotes coach Dave Tippett, whose team squandered a chance to move from 11th place into a wild-card playoff spot in the Western Conference. "They pushed hard and we couldn't push back."
Smith made 26 saves for the Coyotes.
NOTES: The Blues have won four straight in Phoenix. ... Shattenkirk's goal was his first since Jan. 23. ... Bissonnette set a career high with his seventh point of the season. ... Ott, who had been the Sabres' captain before the trade, also was making his St. Louis debut. ... St. Louis' power-play goal was its first in its past 28 chances. ... Phoenix has surrendered the lead in five of its past 11 games.