West Ham endured a torrid first half of the season which ultimately cost Slaven Bilic his job.
The East Londoners now have a season defining run of five games against opposition in the bottom half of the Premier League from their next six.
And Moyes will know if they can gain around 10 points from those matches they will be all but safe this season.
It is a prospect which looked bleak back in December with the Hammers in the bottom three and facing a daunting run of fixtures against Manchester City, Chelsea and Arsenal.
"West Ham with David Moyes, I always felt West Ham were - you never say that about anyone 'too good to go down' - but I think there was probably more in that squad, they'd invested a little bit in the summer," Carragher told MNF.
"And I think with good management they could get more out of them and I think we've seen a lot more of that from David Moyes.
"I'm just pleased to see David Moyes doing well really because he's had it tough the last couple of jobs and I think he's a really good manager what he done at Everton."
A win over Bournemouth on Saturday could send West Ham as high as ninth and just six points off seventh place.
That is then followed by a home game against Crystal Palace, a trip to Brighton a home match against Watford and a trip to Swansea either side of a match against Liverpool.