Berglund tied his career high with his 22nd goal, and the Blues have won 11 of 13 games and have points in 12 of those.
''Just focus on our job, and we want to make the playoffs, but we want to make sure that we're a good team going in,'' Blues coach Mike Yeo said.
St. Louis' Carter Hutton stopped 22 shots and is 7-2-0 in his last nine road starts. He faced only 13 in the first two periods combined, then saved a breakaway attempt by Tobias Rieder early in the third period before Alex Goligoski scored a power-play goal at 14:37.
''He had a few really good sneaky strong saves,'' Yeo said. ''His puck poise and execution and helping our (defense) get out of our zone was real strong, too.''
Perron added an empty-net goal with 19.8 seconds left, his 100th as a Blue.
The Blues moved closer to locking up a playoff spot, falling short of clinching because the Los Angeles Kings beat the Calgary Flames. St. Louis beat Arizona for the 11th straight time and the third time in 11 days.
The Blues beat the Coyotes 4-1 in St. Louis on Monday. They have won nine straight games in Arizona, and during the 11-game winning streak, have outscored the Coyotes 45-14.
Louis Domingue had 40 saves, but Arizona has lost six of seven.
Tarasenko collected a rebound of Jay Bouwmeester's shot and gave the Blues a 2-0 lead at 2:25 of the second period, scoring from an angle for his team-high 36th of the season.
Berglund scored his 22nd goal of the season at 9:16 of the first. He took a pass from Magnus Paajarvi and fired the puck into the net from close range, with Colton Parayko picking up his sixth assist in the past six games against Arizona.
Eighteen-year-old center Clayton Keller assisted on Goligoski's goal. It was the youngster's first career NHL point in his second career game.
''He's a real smart young player. He's real keen on what to do,'' Arizona coach Dave Tippett said of Keller. ''But there's going to be some growing pains. That's the way it is with those young players.''
The Coyotes played their seventh straight game without captain Shane Doan, who is listed as day to day with a lower-body injury. Doan could play as soon as Friday.
NOTES: The Coyotes will honor AHL player Craig Cunningham with a ceremonial pregame puck drop on April 8 when they face the Minnesota Wild in the season finale. Cunningham, who survived a frightening medical emergency prior to the Tucson Roadrunners game on Nov. 19 that resulted in a partial amputation of his left leg, has since recovered and rehabilitated. ... D Kevin Connauton was scratched with an upper-body injury. ... Blues C Paul Stastny missed his fourth straight game with a lower-body injury. C Jori Lehtera (concussion) and F Dmitrij Jaskin (upper-body injury) were also scratched for St. Louis.
Blues: At Colorado on Friday, the first of two games against the Avalanche before the regular season ends.
Coyotes: Host Washington on Friday, the final game against an Eastern Conference opponent this season.
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