Jesse Lingard's first international goal secured a 1-0 victory for the visitors in Amsterdam, the Manchester United player finding the net with a first-time strike in the 59th minute.
England comfortably retained their slender lead through to the final whistle to secure a first win over their opponents since a 4-1 triumph at Euro 96.
"I think that was a good test for us," England boss Southgate told ITV Sport .
"We came away from home and controlled the game. We used the ball really well, especially in the first 60-65 minutes.
"Maybe a little more quality in the final third was required, but overall I'm pleased with the way we played up until that point [the goal], then defensively we looked really solid."
Netherlands came into the friendly fixture on a five-game winning run yet barely troubled goalkeeper Jordan Pickford in Koeman's first game in charge.
Even the early loss of Joe Gomez, who picked up an ankle injury and was replaced by Harry Maguire in the 10th minute, failed to bother a three-man defence that included Kyle Walker in an unfamiliar central role.
"I thought the back three were excellent," said Southgate.
537 - Gareth Southgate’s side have now gone 537 minutes without conceding in all competitions, with their last goal conceded coming against Slovakia in September 2017. Drilled. pic.twitter.com/lZWOxkQaH2— OptaJoe (@OptaJoe) March 23, 2018
"They, along with the goalkeeper, were able to play from the back with real composure. We are going to have to get used to them taking risks in that area of the pitch, because that's the way we want to play.
"But I thought they played with intelligence and composure.
"The defenders and goalkeepers are capable of using the ball well. We were able to work the game out tactically so we could get into areas higher up the pitch.
"There's a lot to work on and we can still get better, there are more things we want to look at on Tuesday, but it was a really good exercise."