Richarlison scored the hosts' opening goal with a towering header from Djibril Sidibe's cross.
Dominic Calvert-Lewin doubled the lead with a smart finish shortly after half-time.
Mason Mount screwed a shot wide and Cesar Azpilicueta brought a save out of Jordan Pickford as Chelsea searched for an equaliser, but the Toffees pounced late as Calvert-Lewin bundled in after goalkeeper Kepa Arrizabalaga's wayward pass to start the celebrations.
Everton attacker Theo Walcott said at the final whistle: "Duncan has come in and the club is his heart. That was the Everton way, we play for each other and Dominic was immense. We wanted intensity and pressure and to get in behind them. We put them under pressure and made them make mistakes. Duncan said go out and play your game and enjoy it. As much as it sounds a cliche, that is what we did."
Calvert-Lewin added: "It was a massive game for us and I have a very close relationship with Duncan. He has stuck by me for the three years I have known him. The most important thing was to go out and give 100%."