With time winding down in the extra period, Mark Scheifle forced a turnover in the neutral zone and the Jets worked the puck into the Colorado zone.
Scheifle sent a pass into a crowded crease, where Wheeler deflected the puck past Semyon Varlamov for the game-winner.
Andrew Ladd scored in regulation for Winnipeg, which won a second straight.
Al Montoya turned aside 33-of-34 shots to claim the victory.
Nathan MacKinnon forced overtime with third-period equalizer for the Avalanche.
Varlamov finished with 35 saves as Colorado dropped a fourth straight.