After winning the opening matches in Minnesota, the A's returned to their home ground in California to win the third match of the series 8-3 in front of a sell-out crowd.
Right-fielder Milton Bradley and third baseman Eric Chavez both hit home runs for Oakland, while shortstop Marco Scutaro added a crucial three-run double in the seventh inning to help the A's to their first playoff series win since 1990.
"I have always been a streak hitter," Chavez said.
"I was ready to hit the strikes when he threw them."
Bradley drove a two-run homer to centrefield in the third inning to give the A's a 4-0 lead, before Twins outfielder Torii Hunter answered with a solo homer in the fourth and Rondell White drove in a run with a sixth-inning single.
That wasn't enough for the Twins however, with Oakland wrapping things up in the seventh inning, when Scutaro doubled in Chavez, Jay Payton and Nick Swisher to seal the match.
"This is the first time we have been able to advance," Chavez added.
"Once you get into a five-game playoff series anything can happen. It has been a while but we were finally able to come through and do it."
Tigers on the brink
Meanwhile, title favourites the New York Yankees are one loss away from being eliminated from the playoffs after losing their second straight match against the Detroit Tigers.
The Yankees and Tigers traveled to Detroit after the teams split the opening matches in the Bronx, with the Tigers prevailing in a 6-0 shut-out to take a 2-1 lead in the five game series.
Veteran left-handed pitcher Kenny Rogers guided Detroit to the win over the New York, using a combination of fastballs, curves and off-speed pitches to frustrate the Yankees' big hitting line-up.
The Gambler: Tigers pitcher
Kenny Rogers held all the aces
41-year-old Rogers pitched 7 2/3 clean innings, allowing just five hits as he took Detroit to the brink of an upset series win.
"I think a lot of people may have had us like David versus Goliath, or whatever, but I think we all felt we had a chance," Rogers said.
"I wanted this win for us as a team, but I wanted it for myself, my family, everything. I wanted to do well and leave nothing out there."
The Tigers took the lead in the third inning, scoring three runs off Yankees starter Randy Johnson, and never looked back
Johnson, who was making his return from a herniated disc in his lower back, had no support from an impotent batting line-up that is renowned for its consistent hitting.
"I thought Randy was fine," Yankees manager Joe Torre said.
"He gave up the three runs in the one inning, but I thought he showed consistency. But offensively, we just couldn't do anything."
The winner of the Yankees-Tigers series will play Oakland in a five match series starting on Tuesday to decide the American League Championship title.