Barkley shone as Chelsea beat Southampton 3-0 on Sunday, netting his first Blues goal.
The England midfielder also had the highest pass completion rate of any player on the pitch making 50 passes or more.
That included an assist to Hazard, with the Belgian netting his seventh Premier League goal of the season.
Chelsea are unbeaten in the league and sit joint-top with Manchester City and Liverpool on 20 points after eight matches.
And though Hazard shone again, the 27-year-old was quick to praise Barkley for his performance with the Belgian believing the former Everton man is destined for a great career.
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“He [Barkley] played a fantastic game,’ Hazard said. “And it is not just because he scored a goal or that he made an assist.
“You can see on the pitch that he is very strong. He had some bad moments in the last year or two years with injury but I think he can reach a fantastic level because he is a fantastic player.
“I remember watching him at Everton. He is still young so I think he is a good player for the future with Chelsea.
“If you give Ross the freedom to play and the confidence on the pitch, he can be one of the best in the world in midfield.
“He is strong, he is powerful, he can dribble, he is clever, he can shoot, he can score and he can make assists.
“If he has players around him to help him to play, he can be one of the best.”
Hazard isn’t the only one to think Barkley is improving with former Manchester United defender Gary Neville also praising the 24-year-old.
“He loves his football, he’s a good lad, but has he been really serious like the really top professionals?” Neville said.
“If you think of Cristiano Ronaldo at the very top of in terms of ‘he looks after himself’.
“He’s getting to that level now, he looks a lot more serious about the game.
“I think he’s getting the rewards for it.”