According to manager Jeff Banister, Dyson was hurt while trying to field a comebacker in the ninth inning of Sunday's 8-7 loss in Seattle. Dyson allowed two runs in the inning, despite not allowing a ball out of the infield, and suffered his third loss.
Dyson is expected to go on a rehab stint in the minors once his hand is healthy.
Banister said right-hander Matt Bush will initially assume the closer duties.
Right-hander Keone Kela was recalled from Triple-A Round Rock.