Everton have got off to an impressive start under their new manager and find themselves in second place behind Manchester City, with 13 points from five games.
The experienced defender admitted that players are slightly scared of the Dutchman, but his methods are reaping rewards.
"There is a fear, definitely, just down to the fact that there have been people who have had good pre-seasons and they haven't started a game yet," Jagielka said.
The former Sheffield United player hailed Koeman's decision-making, as he is not afraid to make tough calls.
"He is not scared of making changes. James (McCarthy) was really unlucky at West Brom, playing out of position and he got brought off before half-time because we needed to do something different tactically," he added.
"He is not afraid of making decisions before and after the game. It is like being back at school with a new headteacher who is trying to be a little bit stricter.
"You only have to look at the jobs he has done. That proves he has not done a bad job.
"At the moment it is going well. I wouldn't say we are playing amazing, but we are getting fantastic results and that breeds confidence".
Can Everton keep up their impressive form for a full season?