Sweeney, though, has stiff competition for a role as an extra outfielder, as fellow non-roster invitee Carlos Quentin is also having a strong spring at the plate, while youngsters Danny Santana and Oswaldo Arcia are both out of Minor League options.
So Sweeney, who doesn't have an out clause in his contract, is still weighing his options if he doesn't make the Opening Day roster.
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"I'm up in the air about it right now," Sweeney said. "But after talking to my agent, I have more than seven years of service time, so they have to let me know a week in advance. So at least I'll have some time. It's not like it'll be the last day of camp and they spring that on me. I'll just have to weigh my options. Like if it comes down to going to Triple-A, I'd have to see the other guys there and whether I'd have potential to even play down there."
Sweeney, 31, signed with the Twins as a Minor League free agent in December because he figured they'd give him the best chance to win a job. He saw how Shane Robinson came in as a Minor League free agent last year and ended up making the team as an extra outfielder, sticking on the roster the entire season.
"I felt like this was a good fit," Sweeney said. "My agent [Larry Reynolds] is Torii Hunter's agent. They have a good relationship with the Twins. And with a young outfield here, it seemed like a good fit to be back in the game."
But after taking the year off, Sweeney wasn't sure how he'd fare this spring, especially at the plate. He started working out earlier than usual this offseason to get ready, but even he didn't expect he'd have this kind of success so far.
"I've been surprised," he said. "That's the last thing I thought that would come facing live pitching. I wasn't really worried about defense, my arm or hitting in BP. It was more about my timing at the plate. But I feel like I'm doing all right. Obviously, you have to come in here and put up good numbers in Spring Training, but I have to show them I'm healthy and can still play."
The Twins have been impressed with his play so far, and he offers more versatility than Arcia or Quentin because he can handle center field. But it's still an uphill battle, because the Twins would have to risk losing Arcia or Santana to waivers, because they're out of options.
"He's swung the bat very well," assistant general manager Rob Antony said. "I think he can play the corners better than center. He has played center in the past, but over the course of time, he's gotten bigger and doesn't run as well as he used to. But he's swung the bat and kept himself in the mix. We're looking for that guy who can swing the bat and come off the bench."
For his part, Sweeney is trying not to worry about whether or not he'll make the roster, as he's aware it's ultimately out of his hands. But he also knows it'll be an interesting two weeks leading up to Opening Day and that he'll potentially have a decision to make about his future.
"My whole family right now, everything is like up in the air for the next 10 days," Sweeney said. "So it is what it is. I have nothing to lose."