Stafford also had two assists in helping the Sabres rally from a two-goal deficit, and overcome the loss of co-captain Chris Drury, who left the game in the second period with a deep cut across his forehead.
Drury, who also scored, was hurt after being blind-sided by Ottawa's Chris Neil - a hit that sparked a wild melee that involved all 12 players on the ice.
Thomas Vanek scored twice, including his 31st of the season, while Clarke MacArthur and Dmitri Kalinin also scored in a game in which the banged-up Sabres opened already missing seven regulars.
Dany Heatley scored twice and added an assist, while Jason Spezza had a goal and two assists. Daniel Alfredsson and Mike Fisher had a goal and assist each.
Stafford, taking the fourth shootout attempt, scored when he faked going to his forehand and flipped a backhander past Martin Gerber.
Ryan Miller then sealed the win when he gloved Fisher's attempt. Miller's best save came a shootout attempt earlier when he got his blocker out to foil Mike Comrie, who appeared to have the goalie going the wrong way.