Miller provides the Orioles another solid left-hander among a crop of relievers that has flourished this season, led by Darren O'Day (3-1, 1.10 ERA) and lefty Zach Britton (3-2, 1.65 ERA, 21 saves).
''With the two really talented left-handed relievers at the end of the game, along with Darren O'Day, it gives us a very, very formidable backend of the bullpen,'' Orioles vice president of baseball operations Dan Duquette said.
The 29-year-old Miller has a 2.34 ERA in 50 appearances with the last-place Red Sox, who unloaded several of their veterans Thursday before the end of the non-waiver trade deadline.
Miller has limited left-handed hitters to a .150 batting average and kept righties to a .180 average in 150 plate appearances. He was expected to be in uniform for Baltimore on Friday against Seattle.
''Miller's emergence this year, being able to get out right-handers along with lefties, made him very attractive,'' Duquette said. ''Not just to us. We ended up getting him, but there were a lot of other clubs that were in the market for left-handed relievers. His capabilities and his performance this year really distinguished himself from any other players in that position in the market.''
Duquette liked the Orioles' starting rotation and didn't want to give up a player on the current 25-man roster, so adding Miller appeared to be just the right move.
''This is one area where we could strengthen our club without giving up an asset on the current team,'' Duquette said.
Since becoming a full-time reliever in 2012, Miller has a 2.78 ERA in 140 games while averaging 13.3 strikeouts per nine innings. His contract expires after this season, so there's a chance he won't be with the Orioles beyond this year.
''We know Andy Miller is going to be a free agent,'' Showalter said. ''We think he can impact our ability to make the playoffs. We're willing to take that risk.''
The 21-year-old Rodriguez is 3-7 with a 4.79 ERA in 16 starts with Double-A Bowie this year.
''I'm not excited about losing Eduardo, but you've got to give up quality to get quality,'' Orioles manager Buck Showalter said. ''It's a tribute to our farm system that we could sign and develop someone that brought Andrew.''
Duquette said, ''We hated to give up a young left-handed pitcher. We tried a lot of other combinations short of Eduardo Rodriguez, but in this case it was required to make the deal. ''
Although Miller isn't necessarily a huge name, he comes to the Orioles on a day their division opponents lost star pitchers Jon Lester (Boston to Oakland), John Lackey (Boston to St. Louis) and David Price (Tampa Bay to Detroit).
''I'm hoping we play well enough to see them again,'' Showalter said.