  We define and study a notion of G-dimension for DG-modules over a
non-positively graded commutative noetherian DG-ring $A$. Some criteria for the
finiteness of the G-dimension of a DG-module are given by applying a DG-version
of projective resolution introduced by Minamoto [Israel J. Math. 245 (2021)
409-454]. Moreover, it is proved that the finiteness of G-dimension
characterizes the local Gorenstein property of $A$. Applications go in three
directions. The first is to establish the connection between G-dimensions and
the little finitistic dimensions of &\mathcal{A}&. The second is to
characterize Cohen-Macaulay and Gorenstein DG-rings by the relations between
the class of maximal local-Cohen-Macaulay DG-modules and a special G-class of
DG-modules. The third is to extend the classical Buchwtweiz-Happel Theorem and
its inverse from commutative noetherian local rings to the setting of
commutative noetherian local DG-rings.
