Backes' goal, Lehtera's goal, and two of Schwartz's scores came on power plays. St. Louis scored on all four of its power play opportunities.
"I like the fact we stayed with it right to the end," Blues coach Ken Hitchcock said. "We kept coming. That's the way we have to play. We're a much better team when we don't absorb, we initiate, and I thought we kept initiating the whole game."
Schwartz, who also had an assist, made a shot from the point just before the game ended.
"I didn't think that was going to happen at all," he said. "My teammates wanted me to stay on the ice and try to get it. I never had a hat trick, so it was pretty special."
Allen had a big game two nights after regular Blues goalie Brian Elliott shut out Los Angeles in a 1-0 shootout loss.
"We've had really good goaltending every game we've played so far this year," Hitchcock said. "That's really a good sign. They give us a chance to win every night."
Beleaguered goalie Mike Smith had 30 saves for Arizona and has allowed 15 goals in his three games.
"We didn't move quick enough, move the puck quick enough, check quick enough, all the above," Arizona coach Dave Tippett said. "It was a test against a real good team and we did not fare very well."
Allen, the AHL's top goalie last season, made his first NHL start since April 21, 2013.
Both teams lost key players to injury.
St. Louis' Paul Stastny, who had four points in three games before Saturday, left with an upper-body injury early in the second period. He was holding his left wrist as he left the ice. Hitchcock said Stastny would not play in Sunday night's game at Anaheim.
The Coyotes lost Martin Hanzal to a lower-body injury with 13:21 left in the second period. The Arizona center had to be helped from the ice.
Smith withstood a flurry of St. Louis shots before Arizona's Oliver Eckman-Larsson was called for interference, the game's first penalty, with 2:36 left in the first period. Eighteen seconds into the power play, the puck was free in front of the Coyotes' net when it glanced in off Backes' left skate.
After a video review, the goal was upheld, officials ruling that it was not a distinct kick as required to nullify the goal.
It was Backes' second goal of the young season and the first power-play goal allowed by Arizona. The Blues had a 15-7 advantage in shots in the first period.
Backes had another great chance at a goal in the second period when St. Louis had a 3-on-0 advantage, but his wide open shot missed the net far to the left.
No matter. St. Louis got two later in the period, on goals by Steen and Schwartz. Schwartz netted his second goal in the third period to make it 4-0.
NOTES: Arizona was 10 for 10 on penalty kills before Saturday night. ... The Coyotes have allowed nine first-period goals in their first four games. ... St. Louis has won five straight in Arizona and nine of 11 against the Coyotes. ... Since 2008-09, Backes is the only NHL player with over 300 points and 1,300 hits. ... The Blues played at Los Angeles on Thursday before going to Arizona for Saturday's game. They will head back to Southern California to face the Anaheim Ducks on Sunday.