The two teams were forced to work overtime on a sweltering night in New Jersey until Kopitar broke in alone on Martin Brodeur and flipped the puck under the Devils netminder to get the best-of-seven series off to a dramatic start.
It is the ninth straight playoff road win for the eighth seeded Kings, who can take command of the series with a 10th away win in Game Two on Saturday back at Newark's Prudential Center.
Colin Fraser scored in regulation for the Kings while Anton Volchenkov had the lone goal for the Devils.
Game One produced little in the way of offensive fireworks in the opening period with both teams managing just five shots but it was enough to give Los Angeles a 1-0 lead.
Fraser took a feed from Jordan Nolan and slammed a one-timer past Brodeur, who was making his 200th playoff start in the New Jersey Net.
The Devils managed just four shots in the second frame but one of them was the equaliser, Volchenkov's slap shot glancing off Slava Voynov past Jonathan Quick with just 72 second remaining in the period to bring the capacity crowd to its feet.
The teams flashed their offensive muscles in the third as the goaltending duel that many had been predicting between Brodeur and Quick took centre stage. Both netminders produced sparkling saves to force the contest into overtime.
It marked the first time in a decade that Game One of a Stanley Cup final had gone into extra time.