Midfielder Barkley celebrated his recall to the national team at St Mary’s by setting up Eden Hazard’s opener and then scoring his first Blues goal.
Barkley and in-form team-mate Hazard were the standout performers on the south coast and the latter laid on a stoppage-time third for substitute Alvaro Morata.
The former Everton player was left out of Gareth Southgate’s World Cup squad in the summer following a series of injury issues and won the last of his 22 caps in May 2016.
Head coach Sarri has been impressed with Barkley’s desire to make up for lost time and tipped him for a bright future heading into the international break.
“Today he has played really well. I am really happy with him because from the beginning he has improved, day by day, week by week, match by match,” said the Italian.
“I am very happy for him, for the goal, for the national team.
“He had a very serious injury in the last season, now he’s improving every day but I think the best will come for him.
“Potentially, he is a great player because he has quality, very important qualities from the physical point of view, from the technical point of view, now he’s improving from the tactical point of view.
“I think he will become very important in England and for the England national team.”
Sarri, who replaced compatriot Antonio Conte at Stamford Bridge in the summer, saw his unbeaten start continue on Sunday as Chelsea remain among the Premier League pace-setters.
The division’s top-scorer Hazard slotted in his seventh league strike of the season with half an hour played after Barkley dispossessed Pierre-Emile Hojbjerg and slipped him through.
Barkley then tapped home after Olivier Giroud directed a Willian free-kick back across goal, before Spaniard Morata finished a flowing passing move after being played in by Hazard.
Sarri felt the final goal was the best of his short tenure and epitomised the slick style of play he is trying to implement.
“Something like this, we moved the ball very quick, I think one touch, at the maximum two touches, and then we arrived at the goal so easy,” he said. “I think it’s my football.”
Asked if it was the finest goal scored since he arrived in England, he added: “Sure, of course, because I can see my way of football in this situation.”
Defeat for Southampton continued their miserable recent run in front of their own fans.
Saints have won just one of their last 15 league games at home, meaning Chelsea and Leicester have each won more matches at St Mary’s in that time.
Striker Danny Ings missed a glorious chance to open the scoring for the hosts, while Nathan Redmond rattled the crossbar at 2-0.
Saints manager Mark Hughes admitted his team struggled to impose themselves on the game.
“(It was a) difficult afternoon for us, we didn’t start with the intensity we talked about,” said Hughes.
“We were too passive, allowed them to get comfortable possession too often in our own half.
“We were able to create a good chance with Danny – clearly if we score first then it may well have been a different type of game for us.
“Credit to Chelsea, I thought they were excellent on the day, they look very strong, very assured in possession.
“With the talent they’ve got at the top end of the pitch, it’s difficult to contain them for 90 minutes.”