The win means Chelsea maintain their 100 per cent record after the opening four games of the season.
"I am happy with the result and I am happy with the performance. It was a very difficult game," Sarri told reporters at Stamford Bridge on Saturday.
"They [Bournemouth] are very organised defensively, and defended very well. The first half, the situation was difficult. Maybe we needed to move the ball at another speed, and maybe more movements without the ball.
"But in this kind of match, usually you can win in the last 20-25 minutes. So I was always confident."
When then asked about his perfect opening set of results at Chelsea, Sarri replied: "At the moment I'm very happy with my players. They improved very well during the week and in matches.
"I'm really very happy to have 12 points from four matches, but I always think we can do more. We can improve. Especially in the performance, of course. We can improve."
Chelsea last won the top-flight crown in 2016 in Antonio Conte's first season in charge of the club. Liverpool have been tipped by many to challenge current holders Manchester City for the title this term but, like Jurgen Klopp's side, Chelsea have a 100 per cent winning record so far this term.
Despite that, Sarri chose to play down Chelsea's chances of winning the Premier League this season, saying: "I think that, now, for us it's better to think about the next match.
"The gap at the beginning was 30 points. I think it's very difficult to recover this gap in only one season. I'm very happy with the four matches, but I think only, starting from now, about the next match. Not until the end of the season."