We formulate and demonstrate a new method for quantum 3D calculations of light atom transfer reactions in atom-diatom collisions. The method follows a general scheme of the hyperspherical method, in common with other hyperspherical formulations in the field. The main novelty consists in the hyperspherical elliptic coordinates (xi,eta) used to parametrize the hypersphere. These coordinates have been introduced recently for studying three-body Coulomb systems, and here we apply them to study a system of three atoms. The coordinates are defined and their relation with the Smith-Whitten and Delves coordinates is explored. On account of a big difference between vibrational and rotational excitation energies in molecules, the hyperspherical adiabatic Hamiltonian allows adiabatic separation between xi and eta. This not only greatly facilitates solution of the hyperspherical adiabatic eigenvalue problem, but also provides an approximate classification of the states by a pair of indices (n(xi),n(eta)) representing vibrational and rotational quantum numbers simultaneously for a reagent and a product. Another novel technology exploited here is the Slow/Smooth Variable Discretization (SVD) method. The SVD is used for treating nonadiabatic couplings between the xi and eta motions, as well as between the motions with respect to the hyperradius and the hyperangular variables. The whole scheme is illustrated by calculations for the reaction O(P-3) + HCl-->OH+Cl for zero total angular momentum. It is shown to be very efficient, accurate, and providing a framework of choice for elucidating light atom transfer reaction mechanisms.