It was a thoroughly miserable return to Goodison for West Ham boss Moyes, who has now lost five out of five meetings with Everton since his lengthy spell as their manager ended in 2013.
Wayne Rooney was the star of the show after netting a hat-trick including a wonder strike from inside his own half direct from a Joe Hart clearance.
His first goal came in the 18th minute as he headed in on the follow-up after his penalty was saved by Hart, and he slotted home his second 10 minutes later.
Jordan Pickford then saved Manuel Lanzini's spot-kick just before the hour mark, and after Rooney's outrageous effort from distance, Ashley Williams added a 78th-minute header.
The Scot expressed his dismay at the first-half performance from his 18th-placed side, who are two points adrift of safety and now winless in seven league matches.
"The second half at least I could put my name to the performance - I didn't want to do it for the first, that is for sure," said Moyes, who has overseen a draw at home and two away defeats since succeeding Slaven Bilic.
"Tonight we weren't good in the first half, so we didn't deserve anything.
"But I thought when we needed small things to go for us, they didn't - like Wayne's penalty was saved but it came back to him, and Lanzini's was saved but didn't."