Arsenal went ahead twice in a pulsating first half at Stamford Bridge thanks to Nketiah - who pounced on a weak back pass by Andreas Christensen - and Emile Smith-Rowe, only to be pegged back by equalisers from Timo Werner and Cesar Azpilicueta.
Nketiah restored the visitors' lead in the 57th minute when he seized on another mix-up in Chelsea's defence to make it 3-2, stabbing the ball home after Thiago Silva's interception bounced into his path off Malang Sarr.
Bukayo Saka rounded off the win when he converted a penalty in the 92nd minute after he was adjudged to have been fouled by Azpilicueta.
The win left Arsenal just outside the Champions League qualifying positions in fifth place but level with Spurs on 57 points and only behind them on goal difference.
Chelsea stayed in third place on 62 points having played a game less than both Spurs and Arsenal and still looking on course for their own Champions League spot next season.