As Fedorov - who was taunted by the crowed throughout the night - skated in on Osgood, he fired a high shot that Osgood deflected with his shoulder.
Jaroslav Balastik had scored on Columbus' first shootout chance before Pavel Datsyuk and Jason Williams scored for Detroit. Henrik Zetterberg, who was scheduled to take Detroit's third shot, was not required to shoot with Detroit already up by a goal.
The Red Wings were the last NHL team to be involved in a shootout.
Robert Lang, back from eight games off with a groin injury, scored the first goal of the game just 4:16 in. Brett Lebda sent a hard pass from the point to Lang at the side of the crease, where he redirected it into the net.
Rick Nash tied the game for Columbus with 6:47 left in the period. He was left all alone in the slot to take a pass from the right wing boards. Defenseman Chris Chelios tried to stop him by tripping him on a play that was going to be called a penalty but Nash kept control of the puck and slipped it between Osgood's pads.
The Red Wings regained the lead 3:14 later when Mark Mowers deflected a shot from the point by Lebda.
The Blue Jackets rallied in the third period, when Duvie Westcott scored a power play goal at 8:30. As Lebda skated without a stick, Westcott was able to jump from the blueline into the slot and take a pass from behind the net, one-timing it past Osgood.
Nikolai Zherdev forced overtime, scoring with 4:09 remaining in the third period on a wraparound attempt that caught Osgood looking the wrong way.
Osgood finished the game with 30 saves on 33 shots while Columbus' Pascal Leclaire stopped 35 of the 38 shots he faced.
Detroit was held without a goal on their seven power play chances and held Columbus to one goal on five tries with the man-advantage.
The Red Wings are back in action on Friday when they visit the Chicago Blackhawks.
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