Manchester City's hopes of retaining the Premier League title may look increasingly forlorn but this narrow victory over Aston Villa, courtesy of Carlos Tevez's 12th goal of the season, just about keeps Manchester United in view and leaves their Midlands opponents deep in relegation trouble.

Restored to the starting lineup in place of Sergio Agüero, who picked up a knee injury in training that is likely to keep him out of the FA Cup quarter-final against Barnsley on Saturday, Tevez converted at the end of the first half after Ciaran Clark, the young Villa captain, tried to execute a turn in an area of the pitch where most defenders would have played safe. Clark slipped, Edin Dzeko picked his pocket and, seconds later, Tevez was planting the ball into the Villa net.

Villa never surrendered but from the moment they fell behind it was hard to see them getting anything out of this game. Although Paul Lambert's side had started brightly and came close to taking the lead through Christian Benteke, when the striker's towering header from Charles N'Zogbia's corner was cleared off the line by Tevez, they created little on an evening when Joe Hart was rarely called into action.

At the other end Guzan made several fine saves, in particular towards the end of the first half when City, after a sluggish start, began to play with more purpose. Pablo Zabaleta's left-foot shot was superbly tipped on to the post and Tevez saw a curling effort brilliantly repelled by the Villa keeper.

If anything, City looked stronger when Roberto Mancini was forced to bring on Dzeko to replace Jack Rodwell, who appeared distraught after he pulled up in the first half with a recurrence of the hamstring injury that has plagued him throughout his career.

Rodwell, Mancini confirmed, would be out for up to four weeks. "We are very sorry for Jack," the City manager said. "He played very well against Chelsea [in the previous league game], he was playing very well tonight.

"He's unlucky. But I think it is impossible to resolve his problems in six or seven months because he's had this problem for five or six years. We need maybe more time."

Mancini retains the belief that his side will need to win all of their remaining matches to have any chance of reeling in their neighbours, who are 12 points ahead, although even then it will need United to suffer the sort of dramatic collapse that is difficult to imagine when Sir Alex Ferguson's side are picking up victories so relentlessly. City, on the other hand, have won only two of their past five league matches and started this game just two points ahead of third-placed Tottenham Hotspur. "We want to look only for the first post, we can't look behind," Mancini said.

Although City were by far the more accomplished side here and twice came close to doubling their lead – Tevez had a shot cleared off the line by Matthew Lowton and Yaya Touré hit the post later in the second half – Lambert was left bitterly disappointed that the game was ultimately decided by an error. "The goal was a mistake – I wouldn't have minded if somebody had put one in the top corner and done a bit of skill," he said.

"I don't think it was a foul [by Dzeko]. I think [Clark's] just tried something he maybe shouldn't have tried, probably feeling confident. But he's been excellent for me this season."

For Lambert, the focus now turns to the next two games, which could go a long way to deciding whether Villa, who remain 18th, will survive. On Saturday they travel to Reading, who are one place and one point below them in the table, and the following weekend they host Queens Park Rangers, the Premier League's bottom club.

"The next two games are massive for us, but if we play like that we'll certainly give teams a fright," said Lambert, who will be without Fabian Delph for both fixtures after the midfielder incurred a two-match suspension by picking up his 10th booking of the season late on here.

Villa had a decent shout for a penalty in the closing minutes when Matija Nastasic seemed to impede Benteke by sticking out a leg, although the striker, like Lambert, was more interested in appealing for handball. Mike Dean, the referee, saw no offence and with that Villa's hopes of picking up an unlikely point in their survival battle were extinguished.

"If we keep playing like that we'll stay up," Lambert said.