The contest was tied 1-1 with 7 1/2 minutes left in the final frame when Miettinen's knuckling shot from the right circle handcuffed Sabres goalie Jhonas Enroth and trickled past him for the decisive score.
Ondrej Pavelec turned aside 24-of-25, including all 11 he faced in the third, as the Jets kept their playoff hopes alive with two games remaining.
The victory gave Winnipeg 51 points -- one less than Southeast Division leader Washington heading into Tuesday's match-up between the two squads at Verizon Center.
Aaron Gagnon also found the net for the Jets.
Thomas Vanek scored and Enroth finished with 37 stops for Buffalo.
Swim Week Miami Beach 2014 [PHOTOS]