Former Manchester United midfielder Hargreaves joined City last week in the hope of reviving his career after three injury-ravaged years at Old Trafford.
The 30-year-old is not yet match-fit and has been left out of City's Champions League squad but Mancini believes he will go on to play a big role this season.
Mancini said: "He had a big problem in the last two years but he can recover very well and be a very important player for us.
"I think he can play in maybe two or three weeks and he can always play.
"He is 30 years old, he is young and he was very unlucky.
"If he can recover very well it will not be a problem."
Former England international Hargreaves made just four appearances in his final three seasons at United due to a succession of injury problems.
He was released by United in the summer but posted videos of himself on YouTube in the hope of attracting another club.
West Brom expressed interest but City, who are expected to challenge for the Barclays Premier League title this season, proved a more attractive proposition.
Mancini has been pleased with what he has seen of the player over the past week and is confident he is over his injuries.
The Italian said: "Hargreaves was free, he was without a team.
"We needed a midfielder and we thought this could be good for us. Hargreaves was a fantastic player.
"He is fit. He trained with us this week and he worked last week."
Another player unlikely to come into contention until later in the month is defender Kolo Toure, who is now available again after serving a six-month ban for failing a drugs test.
"It depends," said Mancini. "He is fit but he needs to play. I think in two weeks he can play."
New #24million signing Samir Nasri is expected to make his home debut despite breaking a bone in his hand on international duty for France this week.
England midfielder Gareth Barry is doubtful with an ankle injury but international team-mate Micah Richards if fit despite pulling out of the Euro 2012 qualifier against Wales with a hamstring problem.
Midfielder Nigel de Jong remains on the sidelines with a foot injury and could be out for another fortnight.
Meanwhile, Wigan boss Roberto Martinez may not face the same challenges as Mancini, but he feels the weight of expectation all the same.
While Mancini is expected to oversee a title challenge after another extravagant summer spending spree, Martinez is merely hoping for improvement after last season's narrow escape from relegation.
The clubs are close geographically but poles apart in terms of finances, squad strength and fanbase.
Yet for Martinez, all that is relative.
The Spaniard said: "Every club works with different objectives and different aims and that brings different expectations.
"I think it is fair to say Manchester City, since they won the FA Cup, have gone into a different status.
"They have shown already this season they are title challengers.
"They will be expected to have another fantastic season, but for us we are expecting to have another fantastic season.
"Our aims are completely different but I am extremely proud of the achievements of Wigan Athletic.
"The excitement is to try to improve on that over the next 10 months."
Martinez signed three new players in Shaun Maloney, Albert Crusat and Patrick van Aanholt last week and all three are in contention to make their debuts, but defenders Steven Gohouri and Antolin Alcaraz remain out.
Source: DSG
Source: DSG