Carter's first shot on goalie Martin Brodeur was stopped, but he regained control of the puck behind the net and skated it out to just above the left circle.
The Kings center then unleashed a wrist shot that threaded its way through two defenseman and zipped past Brodeur to his stick side, handing Los Angeles its second straight overtime victory on the Devils' home ice.
Drew Doughty scored an unassisted tally in the first period and Jonathan Quick turned aside 32-of-33 shots for the Kings, who improved to 10-0 on the road in the playoffs.
They will seek to take a commanding lead in the series when it moves to Staples Center for Monday's Game 3.
Ryan Carter scored on a redirected shot in the third period for New Jersey.
Brodeur finished with 30 saves.