The 2010 World Cup runners-up were sunk by goals from Klose, Thomas Muller and Mesut Ozil in front of a sellout crowd of 51,500 on a chilly night in the German port city.
Muller put Germany, who finished third at the World Cup, in front after a lightning fast attack in the 15th minute that epitomised the home side's domination of the first half.
The striker is now five goals short of Gerd Muller's record tally of 68 for Germany.
"All three goals were set up beautifully," said Joachim Low after his 75th match as Germany coach and his 52nd win.
"It was a richly deserved victory, there's no doubt about that," added the 51-year-old. "The Dutch defence was overwhelmed."
Netherlands coach Bert van Marwijk said he was "hurt" by the defeat.
"That was an embarrassing match for us, to get beaten 3-0 by our neighbours and so deservedly," he said. "But it's good it happened to us before the Euros and not during it."
"If all our players are fit we've got no reason to be afraid," he said.
