The Toffees were playing their first match since the death of chairman Bill Kenwright on Monday at the age of 78.
It had been emotional week, too, for Hammers manager David Moyes, who formed a close friendship with Kenwright during his 11-year spell in charge at Goodison Park.
But while West Ham slumped to a third defeat in eight days following reverses at Aston Villa and in the Europa League at Olympiacos, Everton were able to put some more breathing space between themselves and the bottom three with a second away win of the season.
Calvert-Lewin’s goal was his 50th in the Premier League for Everton, joining Romelu Lukaku, Duncan Ferguson and Tim Cahill in reaching the half-century.
It came six minutes into the second half after a first half that will not live long in the memory.
Calvert-Lewin had his first chance from Vitalii Mykolenko’s cross but the striker headed over.
A torturous half was summed up by the audible sigh which echoed around the London Stadium when five minutes of stoppage time was announced.
The game desperately needed a goal and it almost came through Bowen, who got on the end of a James Ward-Prowse free-kick but guided his header too high.
Instead the goal arrived at the other end after Jarrad Branthwaite won the ball back for Everton in midfield and fed Calvert-Lewin.
It was Calvert-Lewin’s sixth goal in all competitions against the Hammers, the most he has scored against one single club.
Everton almost doubled the lead when Zouma and Aguerd got in another tangle but Areola got down well to tip Aboulaye Doucoure’s shot wide.
The closest West Ham came to an equaliser was a Said Benrahma volley which Pickford kept out at his near post to secure the points.