Barkley seems to have been given a new lease of life under new Chelsea FC boss Maurizio Sarri and he featured from the start in his nine Premier League appearance of the season at Turf Moor.
Alvaro Morata gave the visitors the lead in the 22nd minute, before Barkley made it 2-0 in the 57th minute with a superb goal.
Willian then added a third in the 62nd minute, before Ruben Loftus-Cheek netted in the 92nd minute to help Chelsea FC move up into second place in the table.
Chelsea FC, who were without the injured Eden Hazard on Sunday, are currently two points behind leaders Liverpool FC as they look to try and challenge for the title this season.
And Barkley seems to feel that the Blues are establishing themselves as real title contenders this term.
Barkley told BBC Sport: “I believe we can stay in the title race.”
He then added: “We put in a top drawer performance today. We are growing, we have momentum game by game and we can continue to improve.
“I am made up with how things are going, we are chipping in with the goals.”
Meanwhile, Loftus-Cheek insisted that he will continue to try impressing new boss Sarri.
“It’s been a good week,” said Loftus-Cheek, speaking to BBC Sport. “We need to keep it going. I have to be ready to play whenever, I came off the bench unexpectedly.
“I’ll carry on learning on the training pitch and getting better.”
Chelsea FC will host Frank Lampard’s Derby County in the fourth round of the League Cup on Wednesday night.
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