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."
City, who have begun their league campaign with three successive wins, return to action after the international break as another unbeaten side, Wigan, visit the Etihad Stadium.
City have an exciting week ahead as they make their Champions League bow against Napoli on Wednesday but Mancini is not yet looking beyond the weekend.
He said: "It will be a very hard month because we play every three days. They are all important games.
"I need to think one game at a time and I want to think about Wigan.
"Wigan have a fantastic manager, they play football very well. They want to attack and score.
"This is difficult - every game we have played against Wigan we have won but we have had a problem.
"We will have to have good concentration and a good attitude.
"It is important to have the same attitude we had in the first three games."
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.
Mancini again had to answer questions about controversial striker Mario Balotelli's off-field behaviour as he held his regular pre-match press conference this morning.
The Italian has rarely been far from the headlines since joining City last year and has had another interesting week while back in his homeland.
The £24million man was mobbed while visiting a prison in Florence, called as a witness in a mafia trial and accused of using his iPad while on the Italy bench during a match against the Faroe Islands.
Mancini, who clashed with Balotelli during the pre-season tour of the United States, believes everything concerning the player is blown out of proportion.
He said: "I'm disappointed because everything Mario does is a big situation.
"I think he played well for Italy on Tuesday. This is important.
"The other situations are not a problem.
"Mario is 21 years old. He can change in two months, three months, six months."