This win is playing a crucial role in preserving his career in Westham. If Westham manages to acquire more points, it will undoubtedly contribute to securing Moyes's position.
I've been lurking around their subreddit and others' social media, it seems that some of them still want Moyes out for some regeneration and change. The argument mostly comes from the fact that even though they are winning, they never rely on a good system and rely on key players instead, which is not sustainable or a top 6 finish or more. Not sure what the sentiment is on the local supporters though, but if they got another winless run things could get ugly quickly I bet. Especially if other clubs are interested in your important players like Bowen, there is no guarantee you can find a replacement or other players can step up to fill his shoes.
I think Moyes defended his position recently too, after suffering a big defeat to Arsenal, or their 0-2 loss[1]. He's saying that their recent dip in form is because some of their players are not available. While that is true, he needs to show an alternative since they can't simply wait for their players to be fit and lose 3 in a row whenever Bowen is injured.
[1]
https://talksport.com/football/1753812/david-moyes-hits-back-booing-west-ham-fans/Is Liverpool's victory fortunate? I do not believe that Liverpool is playing lucky; rather, I believe that Liverpool has earned this victory and that Chelsea is embarrassing their supporters.
Chelsea indeed got a bit unlucky at times since their goalscorer missed big chances or made some stupid decisions, but you can't blame luck if your opponent made 9 saves to prevent you put the ball in the net. You could even say the same about Liverpool since they hit the goalpost twice iirc, and Elliot nearly won them in the ET too since the ball is just millimeters from passing the goal line.