Kelly (2-1) took the extra-inning loss against the Astros on Tuesday, allowing an unearned run on one hit over one-third of an inning.
Kelly (1-0) got the win Tuesday against the Rangers, firing a perfect eighth inning.
The Athletics claimed Kelly off waivers from the Guardians on Friday.
Kelly was designated for assignment by the Guardians on Tuesday.
Kelly was optioned to Triple-A Columbus on Sunday.
The Guardians recalled Kelly from Triple-A Columbus on Saturday, Mandy Bell of MLB.com reports.
Cleveland optioned Kelly to Triple-A Columbus on Thursday.
Kelly (back) gave up three runs (two earned) on two hits and one walk while striking out one over one inning of relief in his return from the 15-day injured list in Tuesday's 7-6 loss to the Royals.
Kelly (back) will begin a rehab assignment Thursday with Double-A Akron.
Kelly will be placed on the 15-day injured list Friday with back spasms, Mandy Bell of MLB.com reports.