Everton took the lead after five minutes, Richarlison towering over Andreas Christensen to powerfully head in Djibril Sidibe's right-wing cross.
Calvert-Lewin doubled Everton's lead four minutes into the second half, capitalising on a poor clearance from Kurt Zouma to finish low past Kepa Arrizabalaga.
Mateo Kovacic pulled a goal back three minutes later for Chelsea, firing into the bottom-left corner from 25 yards for his first Premier League goal.
But the Toffees wrapped up victory six minutes from time, Calvert-Lewin latching on to a loose ball in the box to finish underneath Kepa.
Everton move up to 14th place with 17 points, while a third defeat in four league matches keeps Chelsea in fourth on 29 points.
Everton: 15 Dec v Man Utd (A)
Chelsea: 14 Dec v AFC Bournemouth (H)