Rolaids Relief Man Award was created in 1976.
That was Bruce Sutter in 1981.
In the AL, Bill Campbell won in 1976, when he was with the Twins. He also won the next year when he was playing for the Red Sox.
AL - J.J. Putz - Seattle NL - Jose Valverde - Arizona
AL - Billy Koch - Oakland NL - John Smoltz - Atlanta
AL - Todd Jones - Detroit NL - Antonio Alfonseca - Florida
AL - Randy Myers - Baltimore NL - Jeff Shaw - Cincinnati
AL - Dennis Eckersley - Oakland NL - John Franco - Cincinnati
AL - Jim Kern - Texas NL - Bruce Sutter - Chicago
AL - Mariano Rivera - New York NL - Chad Cordero - Washington
AL - Keith Foulke - Oakland NL - Eric Gagne - Los Angeles
AL - Tom Gordon - Boston NL - Trevor Hoffman - San Diego