The Hammers are only off the foot of the table by virtue of goal difference after Daniel Podence hit the only goal of the game to give fellow strugglers Wolves a 1-0 win at Molineux.
After a game of little quality and few chances, West Ham have now taken one point from the last 21 on offer.
It is quite the contrast to the previous two seasons, which saw Moyes lead to club to sixth- and seventh-placed finishes and to the Europa League semi-finals last year.
Now Moyes wants supporters to pay the club back for those successes by backing the team as they continue to struggle, hosting an equally out-of-form Everton next weekend.
“What we’ve given West Ham supporters in the last few years, I hope they can give us back,” he said.
“We are in a difficult period. If they give us back everything the staff and the players have given the club in the last couple of years then I’m sure that will help us.
“You always want the backing of your home support and we’ve always had it. Let’s hope that continues.
“We have probably risen expectations greatly in the last few years, being in Europe, sixth and seventh in the Premier League in the last two years. I can understand disappointment and frustration and maybe they cannot quite understand why it is quite like this.
“It is quite often the journey you have to take when you are trying to climb the tree and trying to get better. Probably with what we have achieved the last couple of years, and avoiding relegation three years ago, building on where we have been means that everybody is a bit disappointed, which they should be.”
Moyes will no doubt come under increasing pressure following the latest setback and he admits he is long enough in the tooth to know his position will be discussed.
“I would never look at myself and say I’ve got credit in the bank,” the former Everton and Manchester United boss added.
“Lots of people are after jobs in this league. I’m not daft, I’ve had enough games in it to understand. You have to do well in it to keep your job. We have to try and do better than we’re doing at the moment.”
While the win takes Wolves out of the bottom three at West Ham’s expense, head coach Julen Lopetegui insists he has no plans to sit and study the table any time soon.
“I didn’t look at the table – not yesterday, not today, not tomorrow,” he said.
“The table is important in the last match, the important thing is to put the energy into helping the players improve and win each match.
“We were fighting a lot for these three points. The second half was very hard for us to keep the rhythm, we are happy for today but it is only three points, it will be a hard and long ride.
“Today I think we deserved it, the character of the players – we celebrate for two hours and then we think about playing Liverpool (in an FA Cup replay).”