Much is known about the geometry of a minimal surface in Euclidean space whose Gauss map takes values on a linear subspace of the quadric hypersurface. We consider minimal surfaces whose Gauss maps take values on rational normal curves. These are the non-degenerate minimal surfaces with smallest possible Gaussian images. We show that the geometry of such a minimal surface may be understood in terms of an auxiliary holomorphic curve on the total space of a line bundle over the Gaussian image. This is related to classical osculation duality. Natural analogues in higher dimensions of Enneper's surface, Henneberg's surface and surfaces with Platonic symmetries are described in terms of algebraic curves.