Cole's second goal of the game came with the Hurricanes holding a two-man advantage. Eric Staal's shot from the slot tipped off Cole's stick and sailed past goalie Michal Neuvirth.
Justin Williams and Noah Babin also scored for Carolina.
Ben Clymer, Mike Green and Nicklas Backstrom had goals for the Capitals, who played without the top line of Alexander Ovechkin, Alexander Semin and Michael Nylander.
Carolina dressed more of its top players than Washington, although captain Hurricanes Rod Brind'Amour wasn't in the lineup.
Washington appeared to be in control after Brent Johnson stopped 12 shots through two periods to help the Capitals to a 2-0 lead. But Babin and Cole scored 52 seconds apart against Neuvirth to get the Hurricanes even early in the third period.
Clymer poked a rebound past Carolina goalie Michael Leighton to give the Capitals a 3-2 edge with 2:03 remaining. But Williams sent the game to overtime when he converted a pass from Ryan Bayda with a minute left in regulation.