City moved into fifth place in the Barclays Premier League, climbing above Liverpool, and will go fourth ahead of Tottenham on Wednesday if they defeat Everton. Roque Santa Cruz and Carlos Tevez scored before half-time at Craven Cottage before Danny Murphy pulled one back from the penalty spot and set up a tense finale.
"It was an important win," Mancini said after the 2-1 victory. "Here, against Fulham, it's difficult. They're a good team and have a good manager."
The former Inter Milan boss added: "It was important for us, but it'll be a fight until the end of the season with the other three teams. Liverpool are a fantastic team, and I think we must fight with them, Aston Villa and Tottenham."
Mancini confirmed England defender Joleon Lescott has suffered a hamstring injury and will have a scan on Monday to determine the extent of the problem. Lescott picked up the problem during the warm-up against Fulham and is the latest player to give national coach Fabio Capello a fitness scare.
"Joleon had a problem before the game in the warm-up, a muscle," said Mancini. "I think he can't play against Everton. At the moment I don't know how long he'll be out. He'll have a scan."
Fulham boss Roy Hodgson, meanwhile, was pleased with his side's response in the second half, even if they could not produce a comeback to match their efforts against Juventus last Thursday.
At 2-1, Stefano Okaka missed a one-on-one opportunity, then referee Lee Probert turned down another penalty appeal when Chris Baird's cross appeared to strike Vincent Kompany's hand.
"I thought the team did remarkably well to come back and had much the better of the second half and came close to an equaliser," said the Fulham boss. "There was a great chance for Okaka that he unfortunately missed, and a penalty appeal that might have led to a second penalty.
"City were much happier to hear the final whistle than we were. We'd have played another 15 minutes, even with a big game on Wednesday."
Source: PA
Source: PA