The Watson trade was announced by the Pirates and Dodgers, who will send minor leaguers Oneil Cruz and Angel German to Pittsburgh in return.
The 32-year-old Watson has a 5-3 record and 3.66 ERA in 47 appearances this season with 35 strikeouts and 14 walks. He will be a free agent after the season.
Cingrani, 28, has a 5.40 ERA over 25 appearances in 23⅓ innings, but has allowed nine home runs.
The Dodgers had only two left-handed relievers with more than 10 innings pitched this season prior to Monday's deadline deals.
Cruz is hitting .240 and German has a 1.91 ERA for Class A Great Lakes.
ESPN Stats & Information and The Associated Press contributed to this report.