The dipole polarizabilities of the ground X(1)Sigma(+) and the lowest-lying A(1)Sigma(+) singlet states of LiH and NaH have been investigated at the ab initio level by using the time-dependent gauge invariant method (TDGI). For the ground-state dipole polarizability of LiH and NaH, and the triplet a excited state of LiH, an alternative method, namely, the coupled cluster with single, double, and a perturbative treatment of the connected triple excitations (CCSD(T)), has been computed for comparison. We found that the long in-plane component alpha(zz) of the excited A(1)Z(+) state of NaH exhibits an unusual calculated negative value, due to the large negative contribution of the X(1)Sigma(+) state. At the TDGI level, the convergence of the calculated electric properties with respect to the number of the spectroscopic states has been carefully investigated. The dipole polarizability components as a function of the diatomic distances up to dissociation have also been computed. The results obtained are particularly relevant to the understanding of the variation of the polarizability for small and large internuclear distances. The properties computed in this work are in very good agreement with the experimental and theoretical data available in the literature. With regard to the a(3)Sigma(+) and alpha(zz) of the excited states, most of the calculated electric properties are new for LiH and NaH.