Carlos Tevez's goal on the stroke of half-time proved decisive as City claimed a narrow win to consolidate their position in second place and keep Villa in the relegation zone.
There are just ten games remaining for City to overhaul their Manchester rivals and Mancini accepts that it will be a tough task, but he is determined to apply as much pressure as possible and hopes United will slip up.
He told Sky Sports: "It was important to stay 12 points behind United because, with ten games to go, you never say never. The championship is not finished.
Asked what his message to the City players was, Mancini replied: "Keep on winning and you never know. This is football. Last year nobody thought it was possible.
"Maybe they can lose four games and we can win every game. We know it will be difficult but we want to try.
"We hope they can lose points before the derby (at Old Trafford on 8th April)."
Mancini was pleased with the way City played against a Villa side he rates highly despite their position in the bottom three.
He said: "I think it was a good performance because Aston Villa are a good team and for us it was important to win this game.
"Villa have young players and probably need more experience, but they play very well and I think they will stay in the Premier League."
City were particularly strong in defence and Mancini was again impressed with the performance of Matija Nastasic in the absence of Vincent Kompany.
He said: "We are playing well. We didn't concede a lot of chances. It was difficult because for me Aston Villa are a good team and it is strange they are near the bottom.
"At this moment we don't have Vinny, who is really important for us, but Matija is playing very well.
"He is young and needs to improve to know the Premier League better, but I am very happy with his performance tonight."
While Mancini says Kompany could return from injury in Saturday's FA Cup tie against Barnsley, Jack Rodwell had to go off with a hamstring problem in the first half at Villa Park and will need to be assessed.