The 1-0 win at Everton was only the third time in their last 19 Premier League matches stretching back to March they had shut out their opponents.
It was a failing which cost them a shot at Champions League qualification as they fell away from the top four in the final couple of months of last season.
After both their defeats so far the Hammers have bounced back with a win and while that is important for Moyes, he is targeting an improvement.
“I don’t know my record at Everton but I don’t think we lost too many back-to-back games,” he said after beating his former club.
“To have success you have to try to do it. Overall you are desperately trying to build on it.
“I have been disappointed we have been conceding goals. It was a good clean sheet and we need more as the season goes on.
“If we are to be successful we have to be better at it. I thought all the players contributed to that clean sheet.”
While he credited the team for the defensive performance Moyes believes centre-back remains a key position.
Kurt Zouma has impressed having arrived from Chelsea for £30million in the summer and has moved ahead of Craig Dawson and Issa Diop in the pecking order but his manager will not allow the 26-year-old to get too comfortable.
“I felt bringing Kurt in we need to step up and I want to raise the bar again so the next change I get, if I can bring in another Kurt Zouma-type, it will continue to bring standards up,” he added.
“He was successful at Chelsea but I want Kurt to feel under pressure with the centre-halves we have got.
“If I had to pick out anyone apart from Declan (Rice), the centre-halves – whatever two I have selected – have been in top form.”
Rice has continued his Euro 2020 form with England in the summer into the new season and Moyes could not be happier, although even the 22-year-old is not exempt from being asked to produce more.
“It would be very hard to say how good he was (against Everton). He was excellent. For me he was head and shoulders above anybody on the pitch,” said the Hammers boss.
“I don’t know if I could speak highly enough of Declan but I’d be saying that about most of the season.
“There have been games where I think he could have stepped up a little bit more but there is big improvement in Declan and obviously that’s good for us.”