"Everything would suggest it's not flexor tendon or ligament, it's more just a strain right in the middle of the forearm," said manager Bob Melvin. "So hopefully that's a good thing. But it's enough to where we felt like we have to DL him."
Cook experienced tightness in his forearm while pitching in the first game of Wednesday's doubleheader against the Mariners, and was removed from the game immediately. He underwent an MRI during Thursday's off-day that revealed the strain.
Rodriguez began the season on the DL while continuing rehab from Tommy John surgery. He has been fantastic since returning to game action for the first time since last March, posting a 0.73 ERA in 11 relief appearances for Sacramento. He did not allow a run over his final 9 1/3 innings.
The right-hander, acquired by the A's with Jed Lowrie from Houston last year, made his Oakland debut Friday night and pitched a scoreless ninth inning in his first Major League game since Oct. 3, 2012, with the Astros.
"Good to see him come all the way back from surgery," said Melvin. "Hard-working kid who really deserves a chance back up here again. We saw him do some good things in Spring Training, and he has been doing well down there, so well-deserved.
"And Joe Savery is another guy that we saw in Spring Training, and that gives you some options, not only against left-handers, but can get some right-handers out, as well."
Savery, who was 4-0 with a 3.86 ERA in 10 relief appearances for the River Cats, allows the A's to keep three lefties in the bullpen, with Drew Pomeranz transferring to the rotation in the wake of Dan Straily's demotion.
The 28-year-old was claimed off waivers by the A's in February, after pitching to a 4.15 ERA in 41 outings spanning three seasons with the Phillies.