Rodriguez will join the team Wednesday in Kansas City, at which point they will make a corresponding move to clear space on the 25-man roster. Jairo Asencio figures to be optioned to Triple-A Norfolk.
The Orioles transferred outfielder Nolan Reimold to the 60-day disabled list to create a spot for Rodriguez on the 40-man roster.
Rodriguez, 31, is 1-1 with a 1.09 ERA and 10 saves in 25 appearances this season, with nine walks and 26 strikeouts in 24 2/3 innings. He's recorded double-digit saves in nine of the last 10 seasons, including a major league record 62 in 2008 with the Los Angeles Angels.
His 304 career saves are third-most among active players, and rank 21st on the all-time list.
"He's got experience in the postseason and he's a proven veteran reliever who can pitch late in games, and he gives us more depth in the bullpen," said executive vice president Dan Duquette.
"He's pitched in the seventh, eighth and ninth, so he's got experience anywhere we ask him to pitch. He's done well in both jobs. He has a really good changeup. And he's throwing well in Milwaukee. He's having a very good year."
Delmonico, 21, was a sixth-round pick in the 2011 First-Year Player Draft out of Farragut High School in Tennessee. He batted .244/.351/.471 in 60 games with Single-A Frederick this season after hitting .249/.351/.411 in 95 games for low Single-A Delmarva in 2012.
Baseball America ranked Delmonico as the Orioles' fourth-best prospect before the season. The Orioles included him on a list of prospects that they deemed expendable and who could bring them a missing piece to the major league club.
"I'm hopeful that we can get a steady stream of prospects into our minor league system to replace the players we trade to help the major league team when it's in a pennant race," Duquette said.