A first half goal from Carlos Tevez was sufficient to earn City a 1-0 win at Aston Villa and cut the gap on leaders Manchester United to 12 points.
But midfielder Rodwell suffered a recurrence of his hamstring injury midway through the opening period and will be sidelined for "three to four weeks".
Mancini said: "We are very sorry for Jack, because he is a good guy, a good player, but he is unlucky. It is his hamstring.
"I think it is impossible to resolve his problem after six or seven months because he's had this problem for five or six years and we need maybe more time.
"How long will he be out? Maybe he will be back after the international break, three or four weeks. It is the same injury.
"I am very sorry because he is a young player. He needs to improve but he has played very well against Chelsea and was playing very well tonight [Monday]."
Mancini repeated his determination that City will not give up on the title without a fight but admits they probably have to win their 10 remaining games.
He said: "It is difficult because we have 30 points to play for and we need to try to win.
"At the end, we will see what happens. In football everything can happen right up to the last second."
Sergio Aguero was also ruled out with a knee injury and will also miss the weekend FA Cup tie with Barnsley but Mancini is hopeful he will be available for the next league clash with Everton.
Villa remain in the bottom three and manager Paul Lambert admitted a mistake by defender Ciaran Clark proved crucial in the build-up to Tevez's decider.
Lambert said: "I thought we were well in the game. The goal is a mistake. I wouldn't have minded if someone had put one in the top corner with a bit of skill.
"But it's a mistake by us. It is easy to be critical of Ciaran. The lad has been excellent all season. He will learn from it, he won't make the same mistake again."
Midfielder Fabian Delph will be suspended for the next two games against Reading and QPR after collecting his 10th booking of the season for handball.
Lambert said: "He's playing extremely well, really well. You are pressing forward, to try and get a goal back and things like that happen."
The Villa boss knows the significance of the next two matches.
He said: "The next two games are massive for us. If we play like that, we will give teams a fright, that's for sure."