Minimization of suitable statistical distances~(between the data and model densities) has proved to be a very useful technique in the field of robust inference. Apart from the class of $φ$-divergences of \cite{a} and \cite{b}, the Bregman divergence (\cite{c}) has been extensively used for this purpose. However, since the data density must have a linear presence in the cross product term of the Bregman divergence involving both the data and model densities, several useful divergences cannot be captured by the usual Bregman form. In this respect, we provide an extension of the ordinary Bregman divergence by considering an exponent of the density function as the argument rather than the density function itself. We demonstrate that many useful divergence families, which are not ordinarily Bregman divergences, can be accommodated within this extended description. Using this formulation, one can develop many new families of divergences which may be useful in robust inference. In particular, through an application of this extension, we propose the new class of the GSB divergence family. We explore the applicability of the minimum GSB divergence estimator in discrete parametric models. Simulation studies as well as conforming real data examples are given to demonstrate the performance of the estimator and to substantiate the theory developed.