Robben twice re-established the visitors' lead in the second half, after a battling Wales -- who have reached their first major tournament since 1958 with a place in France -- had manfully stayed in the encounter despite being outplayed.
Bas Dost headed the Dutch in front after 32 minutes before Joe Ledley equalised and Wales's Emyr Huws then pulled the hosts level again after Robben had danced into the area to score.
Robben then struck the winner in the 81st minute after racing clear to slot calmly into the net.
The Netherlands came into the game still reeling from missing out on next year's European Championship having finished fourth in Group A with four wins from 10 games.
The contrast in mood with Friday's opponents could not have been starker as Wales celebrated their achievement in qualifying for the finals before an enthusiastic Cardiff crowd.
The home fervour was dampened, however, after 32 minutes when Daryl Janmaat lofted a cross to the far post and the towering VfL Wolfsburg striker Dost rose high to direct a downward header past Wales keeper Wayne Hennessey.
ATTACKING FIREPOWER
At that point it looked like it could be a long evening for Wales, deprived of star turns Gareth Bale and Aaron Ramsey, as the Dutch piled forward with plenty of attacking firepower.
Hennessey was forced to turn a shot from Quincy Promes over the bar as the Netherlands turned the screw, until Wales were handed a route back into the match with a penalty when Terence Kongolo was penalised for handball in first-half stoppage time.
Joe Allen's spot kick was parried but Ledley followed up to steer the rebound into the far corner.
The Dutch were back in front within 10 minutes of the restart, however, as Robben produced his trademark finish, dashing up the right flank before cutting on to his favoured left foot and curling the ball powerfully into the net.
Huws typified the battling spirit Wales showed in their Euro qualifying campaign to grab another equaliser after 70 minutes, directing a header past keeper Jasper Cillessen.
Robben, however, had the final word when he ran through a static Welsh defence to finish low past Hennessey.
(Reporting by Toby Davis; Editing by Ken Ferris)
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