Winnipeg's Blake Wheeler was whistled for hooking with 55 seconds remaining in the extra frame, giving the Sabres a 4-on-3 power play. Jason Pominville unleashed a shot from the slot that Vanek tipped in for the game-winner.
Vanek scored during a second-period power play and Derek Roy added a pair of goals for the Sabres, who won their third straight.
Miller -- who sat out Buffalo's two previous wins due to ineffective play -- recovered to make 26-of-31 saves.
Little ended with a pair of goals while Dustin Byfuglien and Glass each had a goal and an assist for Winnipeg, which dropped its third straight to wrap up a seven-game road trip with a 4-3 record.
Ondrej Pavelec made 39 saves for the Jets.