West Ham entered the game looking to bounce back from successive losses to Aston Villa and Olympiacos in the league and Europa Conference League respectively, but Calvert-Lewin's clever finish in the 51st minute proved to be the difference between the two sides.
The result has moved Sean Dyche's side into 15th position in the table, now on 10 points from 10 matches, while West Ham remain ninth, boasting 14 points from their opening 10 games of the campaign.
An inventive pass from Lucas Paqueta almost found Jarrod Bowen inside the Everton box in the ninth minute, but the visitors just about managed to deal with the danger during a strong period for West Ham.
West Ham's first clear opening also involved Paqueta and Bowen, with the former driving into the Everton box before picking out the latter, but his effort was always wide of Jordan Pickford's post.
Everton then had a brilliant chance down the other end in the 25th minute after a mistake from Paqueta, with Jack Harrison driving towards the West Ham box, but he ignored his teammates and instead went for goal, with Hammers goalkeeper Alphonse Areola making a comfortable save.
There was then a flashpoint involving both sets of players after Mohammed Kudus went down to the ground following a challenge with James Tarkowski, with the West Ham attacker booked alongside Pickford, who came out of his goal to get involved.
Harrison sent one high of the West Ham goal in the 34th minute, but there was no breakthrough in the opening half of action, with both sides struggling to show that extra piece of quality in the final third of the field.
Bowen headed a delivery from Aaron Cresswell over the Everton crossbar early in the second half, but it was the Toffes that made the breakthrough in the 51st minute through Calvert-Lewin.
The England international combined with Harrison outside the West Ham box before brilliantly making space and then finding a finish into the bottom corner of the net to spark jubilant scenes in the away end.
Areola made a wonderful save to prevent Abdoulaye Doucoure from scoring a second for Everton just before the hour, with the Toffees looking dangerous in the final third of the field.
Dwight McNeil was next to try his luck in the 65th minute, with his effort needing to be pushed over the crossbar by Areola, as West Ham struggled to stem the flow of traffic.
Calvert-Lewin smashed one off the underside of the crossbar in the 75th minute, although the offside flag had been raised, so it would not have counted, before Pickford made a brilliant save to deny Kudus, but the flag was raised once again.
Pickford then denied Said Benrahma late on, with the Hammers searching for a leveller into the six minutes of additional time, but it was not to be for David Moyes's side, who have now lost their last three games in a row.
Everton, though, have won four of their last six in all competitions, with the team showing huge improvement in recent weeks after a tough start to the campaign.