The midfielder was named as captain in the win over Yeovil.
Everton hosted the League Two side at Goodison Park on Thursday in the second round of the newly christened EFL Cup.
It proved to be a happy evening too as they ran out as comfortable 4-0 winners, courtesy of goals from Aaron Lennon, Ross Barkley and two from Arouna Kone.
It was a particularly special evening for Barkley, who was named as Blues captain for the very first time.
Everton's Ross Barkley celebrates scoring their second goal
And commenting on it, Koeman says that the experience was a good one for the young Toffees star.
“For him, it’s the next step,” he told the club’s official website.
“He is a talent but he also needs [the chance] to show more responsibility for the rest of the team. It was a good experience for a young player.”
Everton's Ross Barkley celebrates scoring their first goal
It will certainly be interesting to see how Barkley develops under Ronald Koeman’s leadership this season.
He managed his best year to date last season in terms of goals and assists, but struggled in the second half of the campaign, something that led to his snub by England at Euro 2016.
Koeman naming him as captain for his game shows the Dutchman has faith in him though and the midfielder will be hoping to get back on track under his leadership.
Everton manager Ronald Koeman before the match