And Reds' midfielder Michael Carrick hopes the club can repeat the treble-winning feat of 1999 with Van Persie.
Arsenal agreed to sell their talismanic 29-year-old Netherlands international striker on Wednesday in a £24m deal.
Redknapp told BBC Sport: "I could see Manchester United winning the league next year, now they've got Van Persie."
The former Spurs manager, who left his job in June, added: "He could be the difference. He really is a great signing."
Carrick added: "If it goes through then it's great news. We welcome him, he had a terrific season last year. He'll just strengthen our squad even more.
"The manager has gone on record as saying that the team in '99, because of the four forwards they had, was one of the most exciting and one of his happiest times. So hopefully we can repeat something like that.
"It's a huge boost. He's a terrific world class player and it's great to bring in players of that ability and give everyone a boost."
Van Persie scored 44 goals in 57 games for club and country last season, including two against United.
He was entering the final season of his contract at Emirates Stadium and stated earlier this summer that he did not want a renewal.
Redknapp predicted success for Van Persie - who joined Arsenal as a 20-year-old in 2004 for £2.75m - at Old Trafford, and sounded an ominous warning to other Premier League sides.
"If he performs like he did last year then him and Rooney will just be an incredible partnership," Redknapp said.