The Hammers skipper says the club should take inspiration from Leicester City and retain belief that they can qualify for the Champions League.
The Hammers are currently eight points off a place in the top four with only 12 points left to play for this season despite their 3-1 win over Watford on Saturday extending their unbeaten league run to nine matches since early February.
At one stage, Slaven Bilic's side were within touching distance of fourth spot and a first-ever participation in the Champions League that would come with it, but four draws in a row coupled with an upturn in form for some of their top-six rivals has widened that gap.
However, Noble insists that the crazy nature of this season, which is like to see last term's relegation battlers Leicester City win the title and defending champions Chelsea miss out on any kind of European qualification, means his side can still make a late surge up the table into the top four.
The Hammers skipper told talkSPORT: "We have had an amazing season. Some of the results we have had have been fantastic and we have given the fans a real good goodbye to Upton Park.
"It is a tough ask [to finish in the top four] with the teams we have got in front of us but this league is so crazy we can go and win the last four games and maybe slip in.
"We are going to look forward to West Brom on Saturday and will go and enjoy it. We have put ourselves in an amazing position where we can go into games and we have got our flair players who can go and enjoy themselves. Hopefully we can win the games we have left and have a right push at the end of the season."
If West Ham can secure a remarkable top-four finish, it would be a sterling end to their 112-year stay at their traditional home ground of Upton Park before they move into the 60,000 Olympic Stadium this summer.