Jones scored on a rebound at 1:09 of OT to give Colorado its fifth win in six games and seventh place in the Western Conference.
Matt Stajan scored the only goal for the Flames and Miikka Kiprusoff made 31 saves.
Calgary has lost three in a row and ended the night two points out of the playoff hunt in the Western Conference.
Stajan scored 1:47 into the game but Stastny tied it at 9:05 of the first period.
The Flames were not penalized in the game but failed to score on three power plays.