Arsenal led twice in the first half, first through Nketiah and then Emile Smith Rowe, only to be quickly hauled back by Chelsea goals from Timo Werner and Cesar Azpilicueta.
But Nketiah added his second early in the second half and a Chelsea team featuring Romelu Lukaku leading the attack this time had no response. Bukayo Saka made sure of the vital three points when converting a stoppage-time penalty.
The victory snaps a three-game losing streak for Arsenal and lifts back level on points with fourth-placed Tottenham and three points ahead of Manchester United, who have played a game more, in sixth.
Despite defeat, Chelsea maintain a five-point cushion in third and have a game in hand on both Arsenal and Tottenham.