The 23-year-old had endured a difficult week due to off-the-field incidents, first allegedly being punched on a night out before finding himself at the centre of a row that resulted in the club banning The Sun newspaper following derogatory comments about him by Kelvin MacKenzie.
However, Barkley was chosen to start against the Clarets and responded by forcing an own goal from Ben Mee en route to a 3-1 win for his side, and Koeman was pleased with what he saw from the midfielder.
When asked whether there was any doubt over selecting Barkley, Koeman told reporters: "No, I think it is the best way to play football. To forget what happened last week.
"Of course I can understand [his celebration] because what happened this week had really a big impact for him. I think he was focused on the football side the rest of the week and also this afternoon.
"I think it is what happens in life. He will learn from what happens. I did myself some stupid things when I was 19, 20, 21. If you learn from mistakes, that is good. Everyone is human."
Everton's win was their eighth in a row in the top flight - their best run since 1990.