The Brewers officially announced the trade, adding that Webb will be assigned to Triple-A.
The 26-year-old Webb made his MLB debut for the Yankees on June 24 with 1.1 scoreless innings against the Texas Rangers. He has a 4.50 ERA with five strikeouts and four walks in six innings this season.
The Brewers, who currently lead the National League Central with a surprising 50-41 record, have had some bullpen problems. They finished the first half ranked 12th in ERA with a 4.08 mark and 22nd with a .749 OPS against.
Jeff Passan of Yahoo Sports noted Cooper could be used as an insurance policy for the Yankees if Greg Bird's ailing foot doesn't heal.
A sixth-round draft pick in 2013, Cooper is hitting .366/.428/.652 with 17 home runs in 75 games for Colorado Springs.