Matt Cullen and Chad Rau scored for the Wild, who also snapped a five-game losing skid at home.
Tim Thomas was tagged with both goals on 29 shots for the Bruins, who have dropped three of their last four.
"We're just shooting it right back into the goaltender. ... That's what happens when things aren't going your way," Bruins head coach Claude Julien said.
After a scoreless first period that saw Backstrom turn away 14 shots and Thomas stop 12, the Wild scored twice in the middle stanza.
"[Backstrom] was outstanding tonight, especially early in the game," Wild head coach Mike Yeo said. "To make those saves early gives us a lot of confidence the rest of the way."
The first goal came at the 10:15 mark by Rau, who was recalled from Houston prior to the game. He skated into the Boston end down the left wing and looked off to his right before blindly snapping a shot on net that beat Thomas to the short side.
Skating on the power play, the Wild took a 2-0 lead as Cullen fanned on his first shot from the slot on the rush, but stayed with the puck and his second attempt from the inside left circle sailed past Thomas with 4:30 left in the frame.
Backstrom had to stand on his head in the third period as he turned aside all 23 shots he faced in the frame for the 26th shutout of his career.
The Wild play in Florida on Thursday...The Bruins continue their six-game road trip in St. Louis on Wednesday...Minnesota has won five of the last six games against Boston...Rau was recalled from Minnesota's American Hockey League affiliate on Sunday after the NHL suspended forward Warren Peters for one game. The suspension stems from a cross check to the head of St. Louis' David Backes late in the third period of a 4-0 loss on Saturday. Peters was issued a major penalty for cross-checking and a game misconduct.