I suppose that we have a fixed contour $\Gamma$ which is a graph over $\partial D$, and you want to show that the minimal graph spanned by $\Gamma$ is area minimizing.

First, by the maximum principle, any (compact) minimal surface spanned by $\Gamma$ has to lie in $D\times R$. Then using Alexandrov's reflection principle, with respect to horizontal planes, I think that one can show that the surface must be a graph over $D$. Then the result should follow from the references you cite.

In short, I am not aware of an explicit reference, but I think what you want should be a straight forward application of maximum principal and Alexandrov's method of moving planes.