St. Louis gave itself some room in the chase for the Presidents' Trophy as the team has 103 points and is two ahead of the Eastern Conference-leading New York Rangers. Also, St. Louis holds a four-point lead over Vancouver for the top spot in the Western Conference.
Phoenix, meanwhile, missed its chance to take over the lead in the Pacific Division. The team is tied with Dallas in points for the division lead, but the Stars have two games in hand. Phoenix sits in seventh in the conference, one ahead of idle Los Angeles.
Alex Pietrangelo and David Backes each posted a goal and an assist for the Blues, who had dropped four of five coming into the contest. Kevin Shattenkirk had three assists in the win and St. Louis killed off all four Phoenix power plays.
"Our penalty killing has been good now for two months and it saved us again," said Blues coach Ken Hitchcock.
Jason LaBarbera gave up all four goals on 29 shots for the Coyotes, who have dropped three of four. They were coming off a shootout loss at San Jose on Saturday.
"It was a pretty quick turnaround," said Coyotes forward Gilbert Brule, "but we just didn't put out a really good effort tonight. That was really unacceptable."
After a scoreless first period, the Blues took a 1-0 lead 33 seconds into the second on the power play when Shattenkirk got the puck to Pietrangelo, who fired a quick shot from the left circle for his 12th of the season.
It was a 2-0 game less than four minutes later when a slapper from Shattenkirk in the high slot was deflected in the slot by Backes.
Patrik Berglund cleaned up a rebound in front of the net with 6:44 to play in the third and Ryan Reaves scored a bit later to account for the final score.
Elliott posted seven saves in the first and third period and six in the second for his 17th career shutout.
Elliott, who posted back-to-back shutouts, is tied with Los Angeles' Jonathan Quick and Henrik Lundqvist of the New York Rangers for the most shutouts in the NHL...St. Louis hosts Nashville on Tuesday...Phoenix hosts San Jose on Thursday...St. Louis has taken all three games against Phoenix this season. The two will meet again in St. Louis on April 6.