The Blues are 7-0-1 since Miller was acquired by trade from Buffalo. St. Louis has won three straight and eight of nine.
Western Conference-leading St. Louis has an NHL-best 101 points. It is the sixth time in franchise history that the Blues have reached 100 points.
Backes has 23 goals this season. Brendan Morrow added a goal, and Jay Bouwmeester had two assists in the Blues' win.
Eric O'Dell had the lone goal for Winnipeg, and Al Montoya made 23 saves.
The Blues won despite the absence of players Vladimir Tarasenko and T.J. Oshie.
Tarasenko is expected to miss six weeks because of a hand injury sustained Saturday against Nashville. Oshie, whose first child was born earlier Monday, should be back for Wednesday's game at Chicago.
Backes broke a 1-1 tie with a power-play goal at 13:09 of the third period when he swatted in a rebound of Ian Cole's shot.
With 1:11 left, Backes was awarded a goal when he broke in on the Jets' empty net and was pulled down by Jacob Trouba.
After a listless first period, St. Louis took a 1-0 lead just 1:44 into the second. After Winnipeg's Devin Setoguchi failed to control a bouncing puck, Derek Roy took it and broke in on left wing. He centered a pass to Morrow on the right side for a shot into the open net.
The Jets took advantage during 4-on-4 play to tie it at 1:55 of the third. Evander Kane skated down left wing with O'Dell on a 2-on-1 break. Kane centered a pass to O'Dell, who faked a shot and then beat Miller with a backhand.
NOTES: St. Louis is 38-1-5 when scoring first and 30-0-4 when leading after two periods. ... Winnipeg is 8-13-5 against the Central Division. ... Blues coach Ken Hitchcock moved into sole possession of third place on the team's career win list with 119.