Jay Payton and Dan Johnson also homered for the A's, whose sixth straight win extended their lead in the AL West to one full game over the idle Angels. The teams begin a three-game series Tuesday night in Anaheim.
"We battled to get in this spot, and we've got to feel proud about what we've accomplished," Swisher said. "We just want to keep it going right now; we just need to keep riding that wave."
Oakland scored all its run on homers. Payton hit a two-run shot in the first inning and Johnson connected with two on in the sixth. The A's have 18 home runs in their last six games.
Kiko Calero (3-1) worked two perfect innings to seal Oakland's first four-game sweep in Baltimore since the A's moved from Kansas City in 1968.
The Orioles had a chance to break a 5-all tie in the eighth, putting runners at the corners with no outs against Justin Duchscherer, but pinch-runner Luis Matos held at third base when Sal Fasano hit into a double play. Brian Roberts followed with a groundout.
The A's were amazed that Matos didn't break for home as soon as Fasano's grounder headed toward shortstop.
"We felt like we were playing with house money when they failed to run home on that ball," third baseman Eric Chavez said. "That was definitely miscommunication on their part. We got lucky, they had a brain cramp on that play."
Orioles interim manager Sam Perlozzo, who spent 14 years as a third base coach, was quick to acknowledge that Matos should have been running on the play.
"That's normally an automatic situation that you go, and obviously we didn't, so I guess that comes back to me," he said. "In that situation, you pretty much want to stay out of the double play no matter where the ball is hit."
Third base coach Rick Dempsey, who moved from first base shortly after Perlozzo took over for Lee Mazzilli on Aug. 4, took the blame.
"It was bad judgment on my part," Dempsey said. "It wasn't (Matos') fault at all. It was 100 percent my fault. He did exactly what I told him to do. I have to live with that."
Said Matos: "The third base coach tells you to stay, you stay. I don't want to look like a bad runner."
Oakland finally took advantage of the mistake in the 12th.
Johnson led off by reaching third base when Eric Byrnes dropped his liner to left, and Swisher followed by hitting a 2-2 pitch from Julio (3-4) over the right-field wall. Marco Scutaro and Jason Kendall both singled before Ellis hit his eighth homer, the fifth in six games.
Three outs later, the A's turned their attention toward the Angels.
"I think it will be fun, two top teams," manager Ken Macha said. "This is where the fun begins."
Jay Gibbons homered and Melvin Mora had two RBI for the Orioles, who have lost nine of 10. After losing two of three to the Angels, Baltimore concluded a 1-6 homestand by being outscored 36-12 in four games against Oakland.
"That was a tough loss," Perlozzo said. "There are two things left to do: You can walk away from here with your tail between your legs, or you can pick it up and go after them tomorrow. We're going to choose to pick it up and go get 'em."
Down 2-0, Baltimore took its first lead of the series with a three-run fifth. After Fasano drew a one-out walk and took third on a single by Roberts, Mora hit a two-run double to left and Javy Lopez chased Oakland starter Kirk Saarloos with a two-out RBI single.
Saarloos, who was bidding to win his fifth straight decision, allowed three runs, five hits and four walks in 4 2/3 innings.
The A's regained the lead in the sixth. Orioles starter Erik Bedard retired 14 straight batters before Scott Hatteberg hit a two-out single and Bobby Kielty walked. Johnson hit Bedard's next pitch into the right-field bleachers, extending his run of games with at least one RBI to eight.
Gibbons hit his 18th homer in the bottom half, and Baltimore made it 5-all in the seventh when Lopez drew a two-out walk and scored on B.J. Surhoff's double.
- Oakland won the season series 7-4.
- It was the second time in nine games that Baltimore scored at least five runs.
- Oakland has scored first in six straight games and outscored the opposition 22-0 over the first three innings during its winning streak.