The move with Moran was made retroactive to Aug. 24. Right-hander Nick Mears and infielder Will Craig, the Pirates' first-round draft selection in 2016, were recalled in corresponding moves.
Craig won a minor league Gold Glove at first base and hit a career-high 23 home runs for Triple-A Indianapolis last season.
Kela has a 4.50 ERA in three appearances this season. He didn't join the Pirates until Aug. 13 after testing positive for COVID-19 in summer camp. The 27-year-old right-hander is 21-11 with a 3.24 ERA and 28 saves in six seasons.
With the Pirates sporting an MLB-worst 7-19 record this season, Kela was expected to be available in trade talks before the Aug. 31 Major League Baseball trade deadline. It is unclear if his injury will change those expectations.