The top flight's bottom club had just one point when Poyet succeeded Paolo Di Canio as manager in October, but the Uruguayan has given Sunderland a fighting chance of avoiding relegation.
After picking up four points from trips to Manchester City and Chelsea over the past week, Sunderland are now just three points from safety, with a game in hand.
Mannone, who has been named the supporters' Player of the Year, is determined to help Sunderland stave off the drop after a rollercoaster season which has also included a Capital One Cup final appearance.
"I'm praying every day that we give our best in the last bit, in the last four games," Mannone told the Sunderland Echo.
"I think we can make it now and it would be something unbelievable to be a Premier League goalkeeper again.
"I hope we can stay up because from where we were, we deserve it, even if we have made some mistakes in games.
"In general, we started from October to November to realise what we could do after the gaffer came in.
"We showed it in the cup run, showed it in some big games. There's four games to go, it's in our hands again and we need to show it again against the small clubs.
"It's been an amazing season, but this club deserves it.
"You see the fans going everywhere and they've been brilliant."
Sunderland's game at home to 18th-placed Cardiff City on Sunday is followed by a trip to Manchester United before the season concludes with matches against West Brom and Swansea at the Stadium of Light.