Filip Gustavsson made 33 saves for the Wild (31-21-5), who finished 4-2-1 on a seven-game homestand. All four of their victories were decided by one goal.
"[The homestand] started off tough," Gustavsson said. "We were out of the wild card spot (in the Western Conference), we had a little dip in our confidence. It didn't feel like the bounces went our way. We had to kind of force it.
"Just get more people to the net. Get more shots there, and then give them less opportunities to score on us. We got one-goal wins, and that's what we need to get it back."
Anze Kopitar scored for the third straight game, and Pheonix Copley made 18 saves for the Kings (32-19-7), who had won four in a row.
It was the start of a five-game road trip for Los Angeles that continues Thursday at the New Jersey Devils.
"We're leaving here without any points, but I thought we played a pretty good game," Kings coach Todd McLellan said. "We played hard, we created a number of chances. … We've just got to find a way to convert, and some nights it doesn't go your way.
"But I didn't think we had a lot of passengers. We had everybody engaged in the game, so we'll chalk it up, we'll move on and try to recover for the next one."
Minnesota forward Adam Beckman had what would have been his first NHL goal taken away on a power play at 9:25 of the first period, but Los Angeles successfully challenged that Hartman entered the zone offside.