An inconsistent run since finishing third at the World Cup, with Saturday's 1-1 draw with Turkey in Euro 2016 qualifying their latest disappointment, had Dutch fans longing for last year and its 5-1 drubbing of the same opponents in Salvador.
However, two goals in the space of three first-half minutes gave Netherlands a second successive win over Spain to help restore some confidence.
De Vrij broke the deadlock after 13 minutes with a well-placed header, before Klaassen netted his first international goal at his home stadium.
Spain were denied a way back into the game when David Silva saw an effort ruled out as they slipped to third friendly defeat in a row.
Hiddink will hope his side can now take some momentum from the win as they look to fight back in Group A given they trail leaders Czech Republic by six points and second-placed Iceland by five.
Having snatched a draw in stoppage time against Turkey, Hiddink's side - which showed five changes - seemed to take momentum forward and started in superb fashion.
Memphis Depay brought the first save out of David de Gea with a curling effort from the edge of the penalty area, but the Manchester United goalkeeper could do nothing about De Vrij's opener, the Lazio defender finding space in the area to guide home Wesley Sneijder's inviting cross after 13 minutes, his third international goal.
The hosts doubled their advantage three minutes later as Klaassen punished some slack Spain defending, the midfielder converting at the second attempt after good work from Depay and Jetro Willems.
Spain – who fielded nine new players from Friday's qualifying win over Ukraine – had flattered to deceive for much of the first half and they found Kenneth Vermeer in fine form when they did pose a threat.
The Feyenoord goalkeeper was alert to deny Gerard Pique after 28 minutes, and he reacted well soon after to clear before Pedro could convert after a jinking run into the penalty area.
Netherlands made another swift start after the break and should have been further ahead after 53 minutes, but Bruno Martins Indi inexplicably failed to convert Sneijder's cross from three yards out.
With both sides utilising their squads, the game lost some momentum but Spain nearly found a way back into the match as substitute Silva saw a 69th minute effort ruled out.
The Manchester City forward was penalised after an Alvaro Morata shot struck him in an offside position before he fired past Vermeer.
Late pressure from the visitors – including a Cesc Fabregas effort - failed to produce the desired effect and Netherlands clung on to what could be a crucial win for Hiddink's future.
