The result leaves the Dutch on 16 points from nine matches, three behind second-placed Sweden, who they face in Amsterdam on Tuesday in their final Group A fixture.
But they will need to make up a 12-goal deficit on the Swedes if they are to make the playoffs.
Sweden have a vastly superior goal difference, and the Netherlands' participation in Russia looks highly unlikely. France lead the pool on 20 points.
Robben’s penalty and another late goal from Memphis Depay secured the win for the visitors, who had led at halftime through Davy Propper's goal. Maksim Valadzko equalised for Belarus in the second period.
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Propper fired the visitors ahead from eight yards on 24 minutes after being teed up by Robben, whose cushioned pass at the back post fell perfectly into his path.
But Belarus pulled level 10 minutes after halftime when Valadzko drilled the ball low past Dutch goalkeeper Jasper Cillessen into the far corner of the net.
The Dutch were awarded a penalty with six minutes remaining when Mikalai Sihnevich was adjudged to have held back substitute Das Bost from a corner and Robben sent his effort low past the keeper.
Depay made the points safe in the third minute of injury-time with a free-kick on the edge of the area that looked to be deflected into the net.
Dutch midfielder Ryan Babel featured in the game from the start, his first cap for his country in six years.
(Reporting by Nick Said,; Editing by Neville Dalton)
