The Rangers placed Winn on the 15-day injured list Monday due to a right rotator cuff strain.
Winn hadn't pitched since Tuesday, when he allowed two runs on five hits over 1.1 innings in a relief appearance against the White Sox. The Rangers haven't revealed the severity of Winn's strain, but he won't be eligible for activation until at least Sept. 12 as a result of the transaction. Texas recalled right-hander Ben Peoples from Triple-A Round Rock to replace Winn in the bullpen.