Balfour emerged as the A's closer this past season, playing a key role in their run to the American League West title.
The 34-year-old right-hander compiled a 3-2 record with a 2.53 ERA and 24 saves over 75 appearances in 2012.
Oakland, meanwhile, declined an option on Drew, 29, who was let go after hitting .250 with five home runs and 16 RBI through 39 for the A's in the past campaign.
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