The Black Cats currently sit second from bottom in the top flight but are just one point away from safety after losing just two of their last seven league matches.
Higginbotham, who played 22 times in one season at the Stadium of Light in 2006-07 before returning to previous club Stoke City, is confident that his former team will avoid the drop under the guidance of Allardyce.
He wrote in the Independent: "[Allardyce] has made a real difference at Sunderland, which is why they will stay in the Premier League.
"Attacks win you titles, but good defences keep you in the league. Allardyce seems to have grasped that better than any other manager in the relegation zone. That is why they signed two defenders in January.
"The hard fact is you cannot score your way out a relegation battle, but you can defend your way out of it. Allardyce’s teams have the mentality that they go on with one point, and they fight to keep hold of that. Thirty-eight clean sheets mean 38 points, as he always says."
Allardyce has a strong pedigree in keeping teams in the Premier League, having steered Bolton Wanderers and then newly-promoted West Ham United to safety in his spells in charge.
He reinforced Sunderland's squad in the January transfer window, signing defenders Lamine Kone and Jan Kirchhoff and bringing in attackers Wahbi Khazri and Dame N'Doye.
He takes his Sunderland players to the home of one of his former clubs this weekend, with the Black Cats travelling to Upton Park to face the Hammers.