Goals from substitute Pedro and Eden Hazard gave the Blues victory over Bournemouth at Stamford Bridge on Saturday.
The result leaves Chelsea level on points with leaders Liverpool, who have a superior goal difference, heading into the first international break of the season.
Sarri succeeded fellow Italian Antonio Conte in the summer after the Blues finished 30 points behind champions Manchester City last season.
"I'm very happy with 12 points in four matches. But I always think we can do more. We can improve," said the former Napoli boss.
"I think now for us it's better to think of the next match. Because the gap at the beginning was 30 points. I think it's very difficult to cover this gap in only one season."
Bournemouth produced a gritty defensive display and missed two glaring chances when the game was goalless.
"Chelsea are a different team this year, a totally different way of playing," said Cherries boss Eddie Howe, whose side won 3-0 at Stamford Bridge last season.
"They'll be a difficult team to play against."
Blues dig deep to continue winning start
Chelsea had to dig deep against a resolute Bournemouth to earn their latest win under Sarri - and once again they left it late to wrap up the points.
Two weeks ago, their winner at home to arsenal did not arrive until the 81st minute. At newcastle last week, it was even later.
Against the Cherries, Hazard's second goal in as many games - which sealed the points - came five minutes from time. Sarri's side had dominated possession but struggled to break down Bournemouth until Pedro struck seven minutes after replacing Willian.
The Spaniard exchanged passes with fellow substitute Olivier Giroud before his deflected finish beat former Blues keeper Asmir Begovic for his third league goal in four games.
Hazard doubled the lead with a quality finish from an angle.
At 0-0, Marcos Alonso was denied by the post before he was further frustrated by a fine save from Begovic.
But victory came eventually - and Chelsea have won their opening four games of the season for the first time since 2014-15, when they won the title.
Bigger tests lie ahead, but the Blues have made a strong start under Sarri.
Positives in defeat for Cherries
This was Bournemouth's first Premier League defeat since losing atsouthampton since april 28. Having stunned Conte Chelsea at Stamford Bridge last season, they were good value for a point until Pedro struck after 72 minutes.
Howe gave Premier League debuts to £25m record signing Jefferson Lerma and Spanish full-back Diego Rico, and his side were hard-working and organised for long spells.
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