Ryan Doumit helped set up Capps' quick memory-clearing opportunity. The designated hitter's sacrifice fly in the eighth against Ryan Cook of Oakland (22-27) drove in Josh Willingham for the go-ahead run.
Doumit called the previous day's 4-3 loss to Detroit a "heartbreaker."
"There's not a bigger competitor on this team than Capps -- no one wants to win more," Doumit said. "There's not a better teammate. When he has a game like this, we're all cheering for him. When he has a game like yesterday, we all feel for him. For him to come in and do the job like he's done so many times before tonight, it was great."
Justin Morneau tied the game with an RBI bloop double before Doumit's at-bat. Morneau's RBI improved his career total to 734, passing Rod Carew (733) for sixth on the Twins' all-time list.
Joe Mauer, who scored on Morneau's hit, started the rally by reaching on a one-out infield single. Mauer stole second when the throw from A's catcher Kurt Suzuki was too high for Jemile Weeks to handle. Willingham reached first on a walk.
The entirety of Minnesota's eighth-inning comeback came against Cook, who hadn't allowed a run all season. Mauer's run was the first surrendered by Cook since Sept. 27, 2011, when he was a member of the D-backs.
"It's not really about giving up the runs," Cook said. "It's about giving up those two runs and losing the game, more than anything. I mean, I guess you're going to give up runs. It's not a secret to anybody. It's unfortunate I gave them up today."
Capps' ninth provided plenty of drama. Alexi Casilla mishandled Collin Cowgill's leadoff grounder and was charged with an error, putting Cowgill on first. Capps induced a flyout from pinch-hitter Seth Smith but then walked Weeks. He threw Coco Crisp out at first after Crisp rifled a ball at the mound.
Manager Ron Gardenhire then visited Capps on the mound, electing to intentionally walk Josh Reddick and bring Gomes up with the bases loaded. Reddick was 2-for-4 earlier, hitting his 14th home run in the first and tripling in the sixth.
The move paid off.
"Reddick is their hottest hitter," Gardenhire said. "He's been killing the ball. I normally don't like telling my closer to walk anybody, but with the way that guy was swinging, we thought that was the right thing to do."
Scott Diamond spotted Oakland an early 2-0 lead after solo home runs in the first and second innings to Reddick and Kila Ka'aihue. Through his first four starts, Diamond gave up a total of two home runs over 25 1/3 innings.
Diamond recovered as the game progressed and went 6 1/3 innings, allowing three runs on nine hits. Though he's only made five starts, his four quality starts are even with Carl Pavano and P.J. Walters for the team lead.
"I'm trying to go as deep into games as I can," Diamond said. "So no matter what, even if I give up runs, I'm going to continue to keep battling and keep going out there trying to get outs."
Prior to Minnesota's bottom-of-the-eighth rally, reliever Jeff Gray walked two in the top of the inning and Alex Burnett surrendered an RBI single to Kurt Suzuki, giving the A's a 4-3 lead.
Oakland starter Travis Blackley locked down Minnesota's lineup early, allowing one run on three hits in five innings. The Twins' offense showed up as soon as Blackley exited, tying the game at 3 with a two-run sixth.
Trevor Plouffe's solo home run in the fifth -- his fourth in nine games -- cut Oakland's lead to 2-1 and was Minnesota's only run against Blackley.