The Hammers inflicted Liverpool’s first loss with a hard-fought 3-2 victory at the London Stadium thanks to an early own-goal from Alisson Becker and further strikes from Pablo Fornals and Kurt Zouma.
The win took West Ham up to third on the table, just three points behind leaders Chelsea.
West Ham missed the Champions League spots by just two points last season but a sixth placed finish afforded them the consolation of Europa League football.
Moyes is aiming higher this time though after recording a fourth straight top flight win on the bounce.
After the game, Moyes was asked by the BBC if his team could challenge for the title, to which he replied: “I want to be really positive all the time, I would like to say that is what we can do. I don’t see that at the moment.
"We can challenge the top four but whoever they are I want to see if we can give us an opportunity, we missed out on Champions League football by two points [last season] so why can we not be there?
"We are not getting carried away but this is feeling good at the moment."
West Ham are away at Wolverhampton Wanderers after the international break.
