His opener was a classic poacher’s goal – poking home after Gylfi Sigurdsson’s deflected strike fortuitously arrived in front of him.
Newcastle improved and after Miguel Almiron struck the post, they equalised when Fabian Schar fired home a volley from Andy Carroll’s nod-down.
But six minutes after setting up the equaliser, Carroll’s mistake led to Everton’s winner.
Theo Walcott dispossessed the striker and played a pass out wide to Richarlison. The Brazilian sped down the right and delivered an excellent low cross for Calvert-Lewin, who slid in and scrambled home.
The Toffees move above the Magpies into 10th, with their opponents – level on 25 points – now down in 11th.