We study a model of neutrino within the framework of minimal extended seesaw (MES), which plays an important role in active and sterile neutrino phenomenology in (3+1) scheme. The $A_4$ flavor symmetry is augmented by additional $Z_4\times Z_3$ symmetry to constraint the Yukawa Lagrangian of the model. We use non-trivial Dirac mass matrix, with broken $\mu -\tau$ symmetry, as the origin of leptonic mixing. Interestingly, such structure of mixing naturally leads to the non-zero reactor mixing angle ${\theta }_{13}$. Non-degenerate mass structure for right-handed neutrino $M_R$ is considered so that we can further extend our study to Leptogenesis. We have also considered three different cases for sterile neutrino mass, $M_S$ to check the viability of this model, within the allowed 3σ bound in this MES framework.