Sutton finished with a pair of goals and an assist for the Thrashers, who won their fourth straight game and kept their playoff hopes alive.
Atlanta, which played its final home game of the season, pulled within one point of Tampa Bay for the eighth and final playoff spot in the NHL Eastern Conference. Tampa Bay was scheduled to host Carolina on Saturday night.
Mellanby and Jim Slater each had a goal and an assist for Atlanta, while Mike Dunham made 23 saves in the victory.
Mariusz Czerkawski, Marco Sturm and Travis Green scored for Boston, which closed its season with five straight losses. Tim Thomas stopped 31 shots in defeat.