Thomas Vanek, Jochen Hecht and Ales Kotalik each had a goal and two assists for the Sabres, who got off to a fast start with four goals in the first period. It was Stafford's first three-goal game, and he also had two assists for a career-high five points.
Colin Stuart scored for Atlanta, which was playing 24 hours after losing in a shootout to Montreal. The loss, coupled with Carolina's win over Edmonton on Friday, dropped the Thrashers out of first place in the Southeast Division.
Buffalo came within two games of tying a franchise-record 12-game winless stretch set in 1991 and matched in 2002.
Daniel Paille also scored for the Sabres, who won for the first time since a 6-5 shootout victory over Philadelphia on Dec. 22.
The Sabres scored their most goals this season. They had scored just 16 times during their 10-game winless stretch.
Atlanta goalie Johan Hedberg allowed four goals on 11 shots before being pulled 12:41 into the game. Kari Lehtonen gave up three more goals on 21 shots through the second period before Hedberg returned for the third.