Brown blasted a sharp-angled one-timer from the bottom of the left circle to beat Wild goalie Alex Stalock 23 seconds into the extra period for his 28th goal and 61st point of the season, a career high.
Kings goalie Jonathan Quick made 23 saves at Staples Center.
With the win, Los Angeles vaulted Anaheim for third place in the Pacific Division, though the Ducks have two games to play while the Kings have just one left. Both teams have already clinched playoff berths.
The Wild will finish third in the Central Division and face the Winnipeg Jets in the first round of the playoffs.
Brown opened the scoring with the game's first shot 55 seconds after puck drop by deflecting Christian Folin's point offering, setting off a crazy game.
Ryan Murphy responded on Minnesota's second shot at the 3:12 mark. After taking a cross-ice pass just inside the blue line, Murphy juked to avoid a checker and sent a shot that deflected off a defender and past Quick, which would be a trend all night.
Matt Dumba tallied 5:02 into the affair to give Minnesota a 2-1 lead when his point shot also ricocheted off a defender.
Then it was the hosts' turn to regain the lead. After Brown's second of the night during a five-on-three power play at 7:42 of the second period evened the count, Tobias Rieder put the Kings ahead 3-2 at 12:57 of the second period with a nifty redirect of Folin's pass.
Daniel Brickley, who signed with the Kings as a free agent out of Minnesota State, collected an assist to make his NHL debut more memorable.
Brown then collected his hat trick with 50 seconds remaining in the second period with a shot that deflected off Wild defenseman Jonas Brodin to make it 4-2.
The Wild roared back, though. At 5:42 of the third period, during a delayed penalty, Dumba sent a cross-ice feed that Eric Staal buried, and then Nate Prosser scored a short-handed goal with 10:36 remaining in regulation to force overtime.
Stalock stopped 23 shots for the Wild.
--Field Level Media
Minnesota | Los Angeles | |
Mikko Koivu 3 | Points | Dustin Brown 4 |
Matt Dumba 1 | Goals | Dustin Brown 4 |
Mikko Koivu 3 | Assists | Christian Folin 2 |
N/A | Power Play Goals | Dustin Brown 1 |
Nate Prosser 1 | Short Handed Goals | N/A |
Alex Stalock .821 | Save Percentage | Jonathan Quick .852 |
Alex Stalock 23 | Saves | Jonathan Quick 23 |
Team | Shots | Goals | Power Play | Penalty Kill | Penalty Mins | Face Offs Won |
Minnesota | 27 | 4 | 0-1 | 5-6 | 18 | 28 |
Los Angeles | 28 | 5 | 1-6 | 1-1 | 8 | 41 |