De La Rosa, who will turn 25 in February, began the year in Triple-A Reno but was promoted in July and spent the remainder of the season with the D-backs, going 2-0 with a 2.95 ERA in 25 appearances. He sported reverse splits, holding right-handers to a .233 average, while left-handers hit .309 off him.
The A's already have a lefty specialist in Fernando Abad and two other southpaws in closer Sean Doolittle and setup man Eric O'Flaherty.
Rodriguez, originally acquired from the Astros in the 2013 Jed Lowrie deal, was expected to be part of this bullpen group next year after the A's opted to tender him a contract this month. He appeared in seven games and held opponents to a .129 average over two stints with the A's this year after missing the 2013 season following Tommy John surgery.
The right-hander also went 3-0 with a 1.97 ERA in 38 appearances for Triple-A Sacramento. The A's have 10 days to trade, release or pass Rodriguez through waivers.
Oakland also announced that right-hander Jorge De Leon, who was designated for assignment last week, has been released.
Jane Lee is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.