The Netherlands needed to win by a seven-goal margin to reach the playoffs but finished level on 19 points, behind Group A winners France, with the Swedes second on goal difference.
The home side started brightly in the Amsterdam Arena with a fired-up Robben converting a 16th-minute penalty and then hammering home a superb left-footed shot five minutes before halftime to raise the faint hopes of the home fans.
The Dutch, however, lacked the necessary creativity as Sweden shut up shop in the second half and the match fizzled out, leaving the Netherlands in third place.
Sweden now enter a two-legged playoff next month against one of the other eight best second-placed European sides.
Tuesday's encounter proved to be the last international for Robben who, after 37 goals in 96 internationals, received a standing ovation as he limped through the final minutes.
After the final whistle, the Swedish players went to their changing room to celebrate, leaving the stage free for Bayern Munich forward Robben to say farewell to the Dutch fans.
“We knew we didn’t have a chance but it was an amazing game. It hasn’t been an easy decision but I am 33 and now must give my full focus to my club,” said Robben.
Missing out on the World Cup caps a remarkable demise for the Netherlands, who were runners-up to Spain at the 2010 tournament in South Africa, and third in Brazil in 2014, with Robben again in the side, in a semi-final defeat by Argentina.
They have now missed out on a second major tournament after also failing in their bid to qualify for last year’s European Championship in France.
The opening goal followed a handball by Swedish defender Victor Lindelof, although Robben miscued a cheeky chip from the penalty spot and was fortunate that goalkeeper Robin Olsen had already committed to dive.
Robben's second goal, however, was a sweeping finish after the ball was laid back to him in the Swedish area. It was to be his last in the famous 'Oranje' national team shirt.
(Reporting by Mark Gleeson; Editing by Toby Davis)
