Off to their best start in the franchise's six-year history, the Wild have won each of their four games by one goal - two by shootout, one in overtime and one with a late comeback.
Minnesota's Manny Fernandez rebounded from a rough first period and stopped Alexander Ovechkin, Alexander Semin and Richard Zednik with tough saves in the shootout.
Kim Johnsson scored two power-play goals for the Wild, who have never lost at home to the Capitals in four meetings.
It was a chippy game, featuring frequent stares, shoves and glares between enforcers Derek Boogaard and Donald Brashear and a whole lot of penalties.
But fans were treated to a thriller at the end, once the five-minute overtime ended without a score.
The nifty manoeuvre by Koivu, who diverted from his usual backhand move and went left to sneak in the winner, ruined a solid performance by goalie Brent Johnson.
With his toe, Johnson kicked away a rebound try by Marian Gaborik late in the third period. The Wild had a four-on-three advantage with 1:49 remaining in regulation, too, and Capitals defenceman Ben Clymer was helpless with a broken stick for a brief portion of that. But Johnson turned away several slap shots in that situation, and in overtime he gloved a riser fired by Nick Schultz with 32 seconds remaining.
Semin and Dainius Zubrus each scored for Washington, which was playing for the first time in five days.
Ovechkin, last season's rookie of the year, had a relatively quiet night, save for the second assist on the goal by Zubrus.
His fellow countryman with the same first name, however, continued his fast start. Semin, who returned to the Capitals this year after spending the past two seasons in the Russian Super League, moved into the NHL lead with his fifth goal - a high shot through man-advantage traffic that gave Washington a 2-1 lead late in the first period.
Semin and Ovechkin have seven of their team's nine goals this year.
Johnson, the backup to fixture Olaf Kolzig, had a strong pre-season for Washington and carried that over to his first start of the regular season.
Johnsson's first score came on an easy tap-in after a sharp cut to the net, fed from a perfect opposite-side pass from Koivu that made the shot nearly impossible to stop.
After scoring only six goals last season with Philadelphia, Johnsson quickly boosted his total with his new team by grabbing a rebound of Gaborik's shot in the second and slapping it into the net from the back of the left circle to tie the game at two.
Notes: Minnesota D Brent Burns, the team's first-round draft pick in 2003, has been a healthy scratch for all four games. ... Wild LW Pascal Dupuis left the ice with a leg injury in the first period and did not return after limping to the training room. ... Captain Chris Clark was credited for his fifth assist on a crisp cross-ice pass to Zubrus earlier that put the Caps on the scoreboard first.