A recently formulated version of the eigenchannel method [R. Szmytkowski, Ann. Phys. (N.Y.) 311, 503 (2004)] is applied to quantum scattering of Schrödinger particles from non-local separable potentials. Eigenchannel vectors and negative cotangents of eigenphase-shifts are introduced as eigensolutions to some weighted matrix spectral problem, without a necessity of prior construction and diagonalization of the scattering matrix. Explicit expressions for the scattering amplitude, the total cross section in terms of the eigenchannel vectors and the eigenphase-shifts are derived. An illustrative example is provided.