The game seemed destined for a shootout when Wild star Marian Gaborik took the puck right in front of the net and fired a shot that Colorado goalie Peter Budaj turned away. But the puck trickled out near the left circle and Burns was there to slam home the game winner.
Mikko Koivu scored two goals and Niklas Backstrom made 21 saves for the Wild, who dealt their Northwest Division rivals a serious setback in their pursuit of the playoffs.
Avalanche rookie Paul Stastny assisted on a first-period goal, giving him a point in his 18th consecutive game and setting the NHL record for the longest point streak by a rookie.
The Avalanche lost for the first time in six games and now face an uphill climb to get into the playoffs. They trail Calgary by nine points for the eighth and final playoff spot with only 13 games to play.
If Colorado can't catch the Flames and squeak into the postseason, it will be the first time Joe Sakic and Co. go home early since 1993-94.
Milan Hejduk and Ian Laperriere scored for the Avalanche, who were dominated in puck possession for most of the game by the Wild.
Minnesota has all but sewn up a playoff spot, leading the ninth-place Avs by 12 points.
Colorado took a 2-1 lead midway through an exciting third period when Laperriere took a centring pass from Ben Guite and beat Backstrom on the stick side.
But the Wild needed just 22 seconds to respond. Pavol Demitra grabbed a loose puck behind the Colorado net and found Koivu to the left of Budaj. Koivu slipped the puck in on the short side, tying the game with just under 10 minutes to play.
Koivu's only other two-goal game this season came at Colorado on Feb. 1, and he has scored seven of his 16 goals this year against the Avalanche.
A couple of impressive penalty-killing units were cracked in the first period.
Koivu put the Wild up 1-0 on a power play midway through the first, ending Colorado's streak of consecutive penalties killed at 17.
The Wild had killed 25 in a row before Hejduk batted home a rebound from Andrew Brunette's shot to tie the game five minutes later.
Stastny got the second assist on that goal, breaking the NHL rookie record set by Teemu Selanne in 1993.
Rookie point streaks seem to run in the Stastny family.
Paul's father, Hall of Famer Peter Stastny, had a point in 16 straight games as a rookie with Quebec in 1980-81. Paul also set a record for getting a point in his 14th straight road game, surpassing his uncle Marian Stastny's 13 straight road games as a rookie with the Nordiques in 1981-82.
Notes: Colorado RW Marek Svatos (groin) was listed as day-to-day but did not dress. ... Wild enforcer Derek Boogaard (back) did not play. ... Wild D Keith Carney had three penalties in the game, which led to two Avalanche goals.