The centre-back flicked home a header late on for a Premier League high 29th set-piece goal – excluding penalties – since Moyes returned to the club for his second spell in charge in December 2019.
It was only the Scot’s third Premier League victory over former Merseyside adversary Rafael Benitez who saw his 100 per cent home record ended, in 16 Premier League matches and his first since 2006 – and all have been at Goodison Park.
“I was annoyed because I thought our play warranted more goals from it or even if we had clear opportunities created from it,” said Moyes.
“But credit to Everton, they defended the box well when we got there and we didn’t show enough in the final third.”
On his side’s prowess from set-pieces, Moyes added: “Nowadays there is a modern trend for set-piece coaches and we want to try to be good.
“We want people who want to head the ball; Craig Dawson was good last year, Kurt Zouma is good, Ogbonna got his goal. We have one or two who can head it.
“I want the team to challenge, I want the team to play wherever we go, get us a victory and I think today we did.
“It is great for West Ham as well as we want to keep fighting and do well in the league and in Europe as well.”
Moyes admitted he had not seen the incident which led to the disputed corner but felt referees’ general interpretation of allowing play to flow more was a good thing.
“I genuinely don’t know the answer but I think there is a bit more contact in the game,” he said.
“We are enjoying it more and the game is a better watch than it was over the last two seasons.
“Overall, the referees are doing a much better job than they have been doing over the last couple of seasons.”
Benitez was deeply unhappy with Michail Antonio’s challenge on goalkeeper Jordan Pickford which led to the corner and ultimately resulted in the winning goal.
“I have a problem because I am from Spain and normally in Europe, the six-yard box is to protect the keeper,” he said.
“When you have a player blocking the keeper in the six-yard area it is a free-kick – and then you can argue whether there is a corner or not.
“You have to protect him. I will not change the perception so the way we concede, after we concede, we have to react and we have to improve.”