In exchange, Chicago obtained 22-year-old third baseman Matt Davidson.
Reed, 24, a third-round 2010 Chisox draft pick, went 5-4 with a 3.79 ERA, 40 saves and 72 strikeouts over 71 1/3 innings in 68 games last season.
The right-hander had his best season during a three-year stint with the White Sox in 2013, establishing new career highs in wins, saves, appearances, innings and strikeouts.
He sports an 8-6 record with a 4.17 ERA and 69 saves in 136 career games.
Davidson, 22, played 31 games for Arizona last season, hitting .237 with six doubles, three home runs and 12 RBI.
He batted .280 with 17 home runs, 32 doubles and 75 RBI in 115 games at Class AAA Reno.