City raced to a ruthless three goal lead within the first 19 minutes, before Aleksander Kolarov completed the scoring late on.
The Chilean believes City can find the points needed to regain top spot in the Premier League if they replicate the attacking football displayed at the Vitality Stadium.
He said: “I’m happy we won again. It’s very important for the team for the future to have the trust to win games in the way we did today.
“Our team must know that we should play in the way we normally do and I think that today we did that.
“We pressed high, we scored goals and playing this way I’m sure that we are going to have the victories that we need to continue to the first position of the table.”
Kevin De Bruyne started for the first time since damaging knee ligaments in January and Pellegrini was pleased the Belgian successfully slotted back into the team with ease.
“I think that Kevin played the same way he was playing before he was injured. He was a very important player for our team and unfortunately he was away for two months,” the Chilean declared.
“He played 55 minutes and I think that he did very well. I hope in the future he can carry on performing like before his injury.”
The manager now turns his attention to Wednesday’s crucial Champions League quarter-final tie at PSG and will assess Joe Hart and Yaya Toure’s fitness in the coming days.
He said: “Everyone asked me if the starting 11 was dependent on what we were going to do on Wednesday but it was not on our mind. Our mind was to be focused on the Bournemouth game.
“Tomorrow we are going to start thinking about the next game and we are going to see what happens with Joe and Yaya.
“It may be difficult for them to be fit for Wednesday, but we will see if they improve in the next few days.”