With both teams waiting on the benches, the start of the game was delayed 30 minutes as former Coyotes captain Keith Tkachuk was inducted into the team's Ring of Honor. The result was a largely tentative first period with few scoring chances and little physical play.
All that changed in the final two periods.
"We're creatures of habit," Elliott said. "It's hard. They got a couple of chances early and you want to make those saves to get your head in the game."
Matt D'Agostini and Jason Arnott also scored for the Blues, who have won six of eight. St. Louis has points in 18 of its last 21 games (14-3-4) since Ken Hitchcock took over as coach after David Payne was fired Nov. 8.
"We really gutted it out," Hitchcock said. "Both teams, especially in the second and third periods, really went right at each other. It was more of a feeling-out process."
"We've been banging our head against the wall trying to figure out why," coach Dave Tippett said. "If you would have told me that by Christmas we'd be a game under .500, I would have been very surprised. That's something that's going to have to change."
After a scoreless first period, Steen gave the Blues a 1-0 lead at 7:12 of the second when he took a long outlet pass from Alex Pietrangelo and beat Jason LaBarbera from the bottom of the left circle.
Doan tied the game 44 seconds later on a long rebound to the inside hash of the left circle, but D'Agostini put the Blues in front to stay with a slap shot from the right circle at 18:25 of the period.
"They had tired people on the ice and we caught them there," Hitchcock said. "But they did the same thing to us."
Arnott made it 3-1 at 1:48 of the third, seconds after Phoenix's Martin Hanzal left the ice after being struck in the face on an inadvertent high stick by David Backes.
"We knew what we were going to get from them," LaBarbera said. "There wasn't a whole lot of room out there and they played a pretty tight game. They got a couple of bounces and it went their way."
Phoenix cut the lead to one when Yandle one-timed a pass from Raffi Torres past Elliott from the bottom of the right circle with 6:21 to go.
LaBarbera was starting in place of Mike Smith, out with an undisclosed lower body injury. ... Tkachuk is the second player who spent the bulk of his career with the franchise in Phoenix to be inducted into the team's Ring of Honor, following defenseman Teppo Numminen. Steen's father, Tomas, also is in the Coyotes' Ring of Honor.
Copyright by STATS LLC and The Associated Press
Arena: Gila River Arena Location: Glendale, Arizona
Referees: Chris Lee, Marcus Vinnerborg Linesmen: Ryan Galloway, Jonny Murray
Attendance: 12,650 (73.9% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
No scoring this period
Alexander Steen (12)
Assist: Alex Pietrangelo
Shane Doan (9)
Assists: Lauri Korpikoski, Michal Rozsival
Matt D'Agostini (7)
Assists: Roman Polak, Evgeny Grachev
Jason Arnott (7)
Assists: Kris Russell, Matt D'Agostini
Keith Yandle (5)
Assists: Raffi Torres, Cal O'Reilly