Barkley came on and played the final 24 minutes as Chelsea FC claimed a 1-0 victory over Videoton in the Europa League on Thursday night at Stamford Bridge.
The midfielder has been settling into life at Chelsea FC after having signed for the Blues from Everton in the January transfer window under Antonio Conte.
He found his first-team opportunities limited last term as he struggled with injury problems, but he is now attempted to force his way into the first team under new boss Sarri.
Barkley has made 10 appearances already in all competitions for Chelsea FC this season – and Blues boss Sarri insists that he has been pleased by what he has seen from the England midfielder.
Speaking at his post-match news conference at Stamford Bridge on Thursday night, Sarri said: “I am very happy for him.
“He had a very serious injury last season. Now he is recovering very well and he is improving match by match.
“He can have a very important future in this club and with the national team.”
Barkley will now be hoping to notch up his third Premier League start of the season for the west London club this weekend.
Chelsea FC are now preparing for a return to Premier League action when they travel to Southampton in the top flight on Sunday afternoon.
The west London side have made a strong start to the new season under their new boss but they are without a win in their last two outings in the Premier League.
Chelsea FC head into this weekend’s round of fixtures in third place in the Premier League table and two points behind leaders Manchester City.
Sarri will be expected to guide Chelsea FC to a top-four finish at the very least this season after they ended up fifth and won the FA Cup under Conte last term.
The former Napoli boss was brought in in July and has already earned plenty of praise for the style of football that he has managed to implement at Stamford Bridge.
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