Williams, who played the whole of his country’s 5-1 win over Belarus in Cardiff, provided the second goal of the Group E contest.
Collecting a pass at the left side of the opposition box, the Liverpool full-back cut inside and squeezed a low finish through the goalkeeper’s grasp and inside the near post.
Wales’ place in the play-offs of the European preliminaries for the World Cup is confirmed.
Van Dijk and the Dutch, meanwhile, must wait until the final matchday of Group G for the chance to book their place at next year’s tournament in Qatar after drawing 2-2 at Montenegro.
The Reds’ No.4 captained the Netherlands and completed the full game in Podgorica, where they led through a pair of Memphis Depay goals but were pegged back by the hosts late on.
Van Dijk and co are two points clear of both Turkey and Norway at the top, and host the latter on Tuesday night.
Elsewhere, Origi was an 84th-minute substitute as already-qualified Belgium defeated Estonia 3-1.