A 1-0 win over the Toffees at Goodison Park extended their unbeaten run on the road to four games at the start of the campaign.
The Hammers have the joint-best away record in the division, with Chelsea and Liverpool the only other sides yet to lose on their travels.
While Moyes naturally wants to keep that form going, he is also hungry for his side to start picking up more points at home, where they’ve won just once in the Premier League this season.
“We’ve got the best away record in the Premier League and what we now want is the best home record in the league at London Stadium, too,” he said.
“We’ve got to try and make sure we get those victories at home as well.
“The players were great today. They played really well. Maybe just not quite as good as I’d have liked in the final third, but our football was so good for long periods. We got chances, but we couldn’t quite those those absolute clear chances.
“I think that’s a bit of credit to Everton as they’ve started the season really well and we have to be aware of how well they’ve done.”
West Ham’s winner against Everton came courtesy of Angelo Ogbonna, who headed home Jarrod Bowen’s free-kick around 15 minutes before full-time.
“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.”
“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.”