Patric Hornqvist began the three-goal rally by scoring 4:11 into the final frame to make it 3-3.
Franson then got the game-winner with 2 1/2 minutes remaining, launching a long wrist shot from near the blue line that found its way past Avalanche goalie Peter Budaj.
The reeling Avalanche, meanwhile, tumbled to their seventh straight defeat.
Kevin Porter scored a goal and an assist, and Budaj made 34 saves in Colorado's latest loss.
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