Wyatt Smith had two goals, and Pierre-Marc Bouchard, Branko Radivojevic and Wes Walz also scored for the Wild, who will meet Anaheim in the first round of the playoffs next week.
Backstrom was coming off consecutive shutouts of Edmonton, but the Blues broke Minnesota's franchise-record scoreless streak at 169 minutes, 20 seconds when Peter Sejna beat the Finnish goalie between the pads with a shallow angle shot in the second period.
Walz came right back on a breakaway, taking a pretty backhand feed from Marian Gaborik and deking Curtis Sanford for a 3-1 lead.
It was an impressive offensive display from the Wild, who rested leading scorers Brian Rolston and Pavol Demitra as they prepare for the second post-season appearance in the franchise's history.
Things seem to be coming together at just the right time for Minnesota, which had a franchise-record, nine-game winning streak last month, but had scuffled a bit after the run was broken on March 27.
But the Wild have won three in a row heading into the postseason and have exhibited many of the elements needed for a deep run - strong goaltending, tight defence and balanced scoring.
It will be another post-season without the Blues, who missed the playoffs for the second straight season after a remarkable run of 25 straight appearances. A wretched start to the season - 7-19-6 - is to blame, but optimism surrounds this young club.
The Blues finished the season strong under new coach Andy Murray, going 27-18-9 with youngsters like Lee Stempniak and Jay McClement stepping in after veterans Keith Tkachuk and Bill Guerin were traded. A win on Saturday would have allowed the Blues to finish above .500.
Their situation could be helped further if a few of their draft picks with Minnesota ties - University of Minnesota defenceman and No. 1 overall pick Erik Johnson and former Warroad High School star T.J. Oshie leave college early and sign with the Blues.
St. Louis will also benefit from the return of goaltender Manny Legace, who played well enough this season to earn a two-year contract extension but missed the last 21 games of the season with an injured right knee.
The Blues certainly could have used Legace against the Wild, who knocked starter Jason Bacashihua out of the game with two goals on eight shots in the first period.
Murray went with Sanford to start the second period, and he made a couple of impressive saves the rest of the way, but allowed three goals on 23 shots.
Notes: Walz's goal was his first in 30 games. ... Former Minnesota Vikings player and Hall of Fame quarterback Warren Moon attended the game. He got a warm ovation upon receiving a Wild sweater with his No. 1 on it in the second period. ... Murray has Minnesota roots from his one-year stint coaching prep power Shattuck-St. Mary's in Faribault in 1998-99.