Barnsley hit the post moment later through Alfie Mawson's fierce drive, while Sam Winnall went close with a header, but Everton finally doubled their lead on the hour when Gerard Deulofeu, picked out by Ross Barkley, rolled the ball into the back of the net.
Having set up one, Barkley scored a thunderbolt of a third, swivelling on the edge of the box before lashing home (71).
Koeman told evertontv: "Maybe on one or two occasions we were a little bit lucky to not concede a goal. But the players showed good work-rate, played good football, created a lot of opportunities and we scored three goals, which is good and what you like to see.
"I'm happy with most things I've seen so far. Of course, we can do things better - we can communicate more than we did today. But you have to understand that it's the third week of pre-season and we'll make steps."