The Oilers won their second in a row and became the last team from the Western Conference to reach 20 victories for the season. New York had a four-game winning streak halted.
Ryan Smyth put Edmonton in front 2:56 into the game and Derrick Brassard tied it a mere 22 seconds into the middle period.
Yakupov beat Cam Talbot with the game-winning slap shot to reach 11 goals for the season.
Ben Scrivens won for the 10th time in the 2013-14 campaign with four of those victories coming in shutouts. He stopped 35-of-36 shots. Talbot made 29 saves.
There was only one penalty called in the contest, that going to New York's Daniel Carcillo for slashing in the second period.
Both of New York's NHL teams lost at home Thursday to the bottom two clubs in the Western Conference. The Islanders fell to Calgary, which is the only team in front of Edmonton in the West.