Jonathan Quick appeared to have brought the Kings their first win of the season, but surrendered the goal to Yakupov on a tip-in as time was running out. Yakupov played in the third game of his NHL career.
Gagner then scored 3:01 into the extra period to keep the defending Stanley Cup champions winless in three games.
The Kings picked up their first point of the campaign with the overtime loss, leaving Washington as the only team in the NHL without a point this season. Carolina and Philadelphia both won for the first time on Thursday.
Gagner's goal came on the power play after the Kings had picked up a bench penalty for having too many men on the ice.
The Los Angeles goal came from Jeff Carter on a backhander in the first period. Edmonton's Devan Dubnyk stopped the other 30 shots he saw to get the win.
Quick finished with 36 saves.