The 32-year-old Hendriks posted an 8-3 record and recorded 38 saves in 44 chances for the White Sox this past season. He also had a 2.54 ERA and 113 strikeouts across 71 innings during his first season in Chicago.
Hader, who also won the award in 2018 and 2019, was 4-2 with a career-best 1.23 ERA in 2021. He had 34 saves in 35 opportunities and struck out 102 in 58 2/3 innings.
The AL runner-ups were Los Angeles Angels right-hander Raisel Iglesias and Toronto Blue Jays right-hander Jordan Romano. The other finalists for the NL award were Los Angeles Dodgers right-hander Kenley Jansen and Mark Melancon of the San Diego Padres.
Voting for the awards was based on regular-season performance and was conducted among seven former MLB relievers: Rivera, Hoffman, Dennis Eckersley, Rollie Fingers, Lee Smith, Billy Wagner and John Franco.