The win put the Blues one point ahead of the north-western pair prior to their own clash on Sunday at Anfield, having played one more game.
But according to Sarri, it is still far too early to talk of Chelsea as title hopefuls.
"It's the same that I thought one week ago. I think Manchester City and Liverpool are a step forward," Sarri told reporters.
"Now we are trying to cover the gap and we are working on that. I think it will be very difficult to cover so large a gap in one season.
"I think sometimes, not always, we can do better, they had two or three opportunities. We need to be in control of the matches."
Sarri's assessment was not shared by Southampton boss Mark Hughes, however, whose side sit two points above the relegation places.
"They will always be contenders, we've been beaten 3-0 by them today, I’m not going to change my mind," Hughes said.
"They're an outstanding team."
The Italian also had glowing praise for Barkley, who has gone from strength to strength under his watchful eye this season.
The former Everton man also claimed an assist, setting up Hazard for the opener, and Sarri – who has shown faith in Barkley since taking over at Stamford Bridge – insists the former Everton midfielder has the potential to become a key figure for Chelsea and England.
"Potentially he is a great player," Sarri told a news conference.
"He has quality, very important ones in the physical and technical [aspects], now he is improving in the tactical point of view.
"He can be a very important player in England and for the national team.
1 – Ross Barkley is the first Englishman to both score and assist in a Premier League game for Chelsea since Frank Lampard in December 2013 vs Sunderland. Rise. pic.twitter.com/AGOrZgYAAb— OptaJoe (@OptaJoe) October 7, 2018
"I am very happy with the goal and assist for him. I am very happy with him, he is improving all the time.
"I spoke to him from the first day that he has great physical and technical quality. Now he has improved."