The City manager believes United's £24 million (Dh142.5m) signing can make the difference in the title race and rued his club's inability to tie up a deal for the Dutchman.
Van Persie had claimed he listened to his heart when opting for United in August but Mancini insisted City almost signed him earlier in the summer.
"We were very close. He took his decision but three or four months before we were there," he said, bringing his forefinger and thumb together to indicate how near they came to completing a deal.
"We wanted Van Persie because we knew he could be an important player," he added. "He is totally different from our other strikers. We were very close. We were sure it was good for us but it did not happen."
Van Persie scored an injury-time winner for United against City last month, a reason why Sir Alex Ferguson's side have a seven-point lead in the title race.
And the 29-year-old Dutch national's 14 league goals have been worth 19 points to United already this season, and Mancini added: "It is clear he has changed the situation. In this moment, he is the difference."
While Van Persie was the Premier League's most prolific player last season, getting 30 goals for Arsenal, City nonetheless were the 93-goal top scorers. And Mancini was adamant he was not lamenting the one that got away, rather than being critical of the four forwards he does have.
He explained: "I am happy, very happy. Last year we were the team that scored the most goals in the Premier League. But when you can improve your team with top players it is important. With Van Persie we could also play with three strikers sometimes."
While United's biggest summer buy has made a major difference, City's summer signings, with the exception of Matija Nastasic, have been criticised for their lack of impact.
Mancini added: "I believe that these players can do better. [Javi] Garcia started very well and then the week after he had an injury and he was out for 40 days. Now he starts to play again, Garcia will be important for the future. Scott [Sinclair] is my fault because I didn't give him a big chance to play."
Jack Rodwell and Maicon, the other costliest summer additions, have spent much of their City careers on the treatment table and injuries are a reason why Mancini ruled out letting players leave in January.
Joleon Lescott, who has lost his place to Nastasic, may be eyeing the exit but Mancini said: "I can understand that Joleon wants to play because he is an England player and it is difficult for him. But we can't sell Joleon because we don't have other players."
The City manager will be without the injured Rodwell, Maicon, Micah Richards and Aleksandar Kolarov and the ill Mario Balotelli for today's game against Stoke City while Yaya and Kolo Toure could play their last games before leaving for the African Cup of Nations.
Mancini is unhappy his Ivorian players will miss Saturday's FA Cup third-round tie against Watford because of a meeting with their national team, rather than an actual match, and is perplexed that the English Football Association have written to him ask him to explain his comments about referee Kevin Friend in the defeat to Sunderland
"It was a joke when I said the referee had a big lunch for Christmas," said the Italian, before quipping: "I didn't receive a letter. The post is too slow in this country."