Sunderland are currently just one point away from safety.
Keys has made no secret of his admiration for Allardyce, and has gone on record to state his opinion that West Ham United should not have sacked the 61-year-old.
However, the pundit feels that the Hammers’ loss has been Sunderland’s gain, as he is convinced the Black Cats will avoid the drop.
“Big Sam has given Sunderland a real chance of staying up,” Keys wrote.
“Allardyce knows what he's doing and I've never doubted he would keep his team in the league.”
Sunderland have looked in serious danger of going down at a number of stages throughout the season so far, but they have moved to within a point of safety after beating Manchester United in their most recent outing.
Allardyce’s men were much improved against the Red Devils, and they managed to pick up a 2-1 victory after scoring through Wahbi Khazri and Lamine Kone.
Khazri and Kone were both brought to Sunderland in the January transfer window, alongside Jan Kirchhoff and Dame N’Doye, and all four players have made good initial impressions at the Stadium of Light.
The Black Cats have improved considerably in their last three matches, where they have collected four points, and they are now within touching distance of Norwich City in 17th place.