With Sam Allardyce watching from the stands ahead of his expected appointment as Everton manager, Rooney found the net for the first time in five games by heading in on the rebound in the 18th minute after his penalty was saved by Joe Hart.
Rooney struck again at the near post and piled more misery on Moyes by capitalising on a scrambled clearance from Hart and blasting the ball past his former England team mate from inside his own half.
Ashley Williams completed the rout by heading home from a corner after conceding a penalty for a foul on Diafra Sakho when the score was 2-0, but Jordan Pickford came to the rescue by turning away Manuel Lanzini's spot-kick.
West Ham had come out fighting in the second half after a meek start and came within inches of pulling a goal back when Aaron Creswell hit the crossbar before Lanzini spurned the ideal opportunity to get them back into the game.
They have taken one point from three games since Moyes took over from Slaven Bilic.
"Not good in the first half but a massive improvement at half-time. We didn't deserve that final score. Mistakes cost us," Moyes told reporters.
"The missed penalty [at 2-0] was the turning point. It looked too difficult a challenge once we didn’t score it. We never really got at it from the start. We allowed them to get a bit of confidence."
Everton had come into the game reeling from a 5-1 thrashing at home by Atalanta in the Europa League and a 4-1 loss at Southampton but appeared lifted that Allardyce was in negotiations to take over at the club.
David Unsworth had enjoyed only one win in his previous six games in charge since Ronald Koeman was sacked last month and the former Everton defender praised his side.
"It’s a great night, not for me but for the club," Unsworth told the BBC.
"Every player was committed. I asked them to be men tonight. If you can stand up there and fight in times of trouble, that says a lot.
"Great hat-trick, great display from Wayne. He was captain for a reason. I’m delighted for the club. I took over when we were in the bottom three and we’ve got out of it in the six weeks."
(Reporting by Richard Martin, editing by Ed Osmond)
