Pinch-hitter Delmon Young singled to open the inning and advanced to third when Barton tried for the force at second base on Alexi Casilla's bunt but threw the ball into the outfield. After Denard Span grounded out, Plouffe hit a flyball to right field off Brian Fuentes (1-4).
"That's baseball," Plouffe said. "Most teams will capitalize on other people's mistakes. They did when I made the error and we did when they made the error."
Barton came within 10 feet of tying the game in the 10th but Michael Cuddyer made the catch on the warning track, ending the game.
"I was just trying to figure out if I was going to catch it in the air or let it drop," Barton said of his error. "I had him at second and just threw it away. Bad throw."
"Nathan made some great pitches and that's what we're looking for him to do," Twins manager Ron Gardenhire said. "We have been keeping him away from situations to let him get his feet wet but it's time to push the envelope. We need him to make big pitches in the seventh and eighth innings."
Justin Morneau also drove in a run for the Twins, who have won two straight following a nine-game losing streak.
"As easy as you lose games you can win games," Plouffe said.
"When you play that close, that late, you have to do everything right to win," A's manager Bob Geren said. "I never believe one play here or there necessarily wins a game when you play 27 or even 30 outs."
Twins starter Carl Pavano, who was looking for his 100th career win, allowed three runs -- one earned -- on six hits. He struck out two and did not walk a batter. He made his first career start in Oakland and has now started in every AL stadium.
A's starter Brandon McCarthy also lasted seven innings, giving up three runs on nine hits. He struck out one and did not walk a batter.
McCarthy is winless in his last seven starts.
Morneau singled home a run in the first to give the Twins the early edge. Morneau, who has been struggling all season, went 2 for 4 and is hitting .485 (16 for 33) with 13 RBIs over his last 10 games against the A's.
Kouzmanoff singled home two runs in the second to put the A's on top 2-1.
Plouffe, whose throwing error led to the A's first two runs, came back to drive in two runs in the third for a 3-2 Twins lead.
"That's what you want to do," Plouffe said. "I was fortunate enough to come up with runners on second and third and less than two outs twice."
Matsui's RBI single in the sixth tied it.
The Twins hung a No. 3 jersey in their dugout to honor the memory of slugger Harmon Killebrew. The team will continue the practice the rest of the season. Players are also wearing patches in his honor on their uniform sleeves. ... Morneau is a career .313 hitter, with 43 RBIs, against the A's. ... A's RHP Michael Wuertz has not allowed a run in 10 of his 11 appearances this year. ... Twins' LHP Glen Perkins has not allowed an earned run in 18 of his 21 appearances. ... Barton had gone 24 consecutive games without committing an error. He still leads the majors with six errors by a first baseman. ... A's OF Coco Crisp was not with the team because of a family issue.