Moyes needed a little luck for his first goal at London Stadium, with Cheikhou Kouyate's header from a corner finding its way into the back of the net off the back of Danny Simpson.
Moyes needs a minimum of a seven-point return from this five-game run.
"We had to kill the game in the first half, but they scored just before half-time and came back with confidence".
The unavoidable fact of the matter is that the Hammers have won two Premier League games all season. "It has not been the best place for us but I thought the players' performance merited more". For the hosts, Andy Carroll is the favourite to score at 2/1 odds.
"Second half was much more like us".
"They got right behind the players, but I also thought the players gave them something to shout about as well. If we are now saying you can't use your arms for leverage then that isn't in the rules of football".
This game is a tough one to call and most of the bookmakers have put the visitors as slight favourites to win.More news: National Christmas tree shortage skyrockets prices
Albrighton's run into the box was ended by a challenge by Andre Ayew who told the Leicester scorer it was never a penalty.
David Moyes' revealed his desire to see West Ham's supporters get behind the team before Friday's fixture against Leicester.
Marko Arnautovic hasn't enjoyed the best of starts to the season - and the same can be said for his new club West Ham - but the Austrian now insists he is ready to give 'everything' for the club.
For all the positives, West Ham will remain in the bottom three for the next week.
Everton's Europa League mauling by Atalanta doesn't bode well for their trip to Southampton on Sunday, when Arsenal and Manchester City visit Burnley and Huddersfield respectively. An improved performance that offered hope, no doubt, but still far from a solution. People come to the game looking forward to the weekend. After failing to register a shot on target against Manchester City, the Foxes again lacked a clinical approach.
West Ham played out a 1-1 draw with Leicester at the London Stadium in the Premier League on Friday evening.
'After losing a goal after eight minutes we stuck at it. In the second half, we had a real good go at it, maybe in the end we could have nicked it. This lack of ruthlessness could cost them down the line if Claude Puel does not react soon.