The Avalanche were trailing 1-0 in the closing minutes of regulation when goalie Jean-Sebastien Giguere was lifted for an extra attacker.
Gabriel Landeskog gained control of the puck near the net and sent it to the point. Tyson Barrie launched a shot that Parenteau tipped past Cory Schneider for the tying score with 1:47 remaining.
Devils defenseman Andy Greene was penalized for slashing with 42 seconds left in regulation, which carried over into the extra time.
O'Reilly scored on the 4-on-3 advantage 28 seconds later, handing the surging Avalanche a fourth straight win and eighth in 10 games overall.
Giguere finished with 27 stops.
Ryan Carter scored and Schneider made 23 saves for the Devils.