Yaya Toure scored a second-half goal as City beat Stoke 1-0 in a one-sided final in which captain Carlos Tevez and his team-mates outclassed their opponents.
"We won the FA Cup. It is an important trophy," he said. "We need to improve more to do another step (win the title) but it was important to start to win because when you start to win, afterwards everything will be easier."
Mancini also denied City's victory was overshadowed by the fact that Manchester United had picked up the Premier League trophy with the point won in a 1-1 draw against Blackburn minutes before the FA Cup final at Wembley kicked off.
When asked if the cup victory felt devalued, Mancini said: "Absolutely not. In England there are three competitions, the Premier League, Carling Cup and FA Cup.
"We want to try (to match United). We have got to the Champions League, that was our first target and we won the FA Cup. It is very important for this club."
On whether the drinks would be flowing Mancini, whose side play Stoke again in the Premier League on Tuesday, also said: "No, they know that we should play in three days. We have another two games and we want to try if possible to get third position.
"This night all the people in Manchester are happy. Next year we will see. It is important we finish well this season. We can try to play for the top with the other teams but this Premier League is very difficult because Liverpool and Tottenham are also very strong teams."
Mancini singled out goalscorer Toure, whose basic pay packet is reported to be the biggest in the Premier League, for special praise.
Mancini said: "We bought him for this. He is a fantastic player. He scored in the semi-final and the final. All season he has played very well."
Source: PA
Source: PA