We study level-set percolation of the discrete Gaussian free field on the Euclidean lattice in three and more dimensions, equipped with uniformly elliptic random conductances. We prove that this percolation model undergoes a non-trivial phase transition at a deterministic level. For a compact set A in R^d, we study the disconnection event that the level-set of the field below a given level disconnects the discrete blow-up of A from the boundary of an enclosing box, in a strongly percolative regime. We present quenched asymptotic upper and lower bounds on this probability in terms of the homogenized capacity of A. The upper and lower bounds concerning disconnection that we derive are plausibly matching at leading order. In this case, this work shows that conditioning on disconnection leads to an entropic push-down of the field.