Using a Hubbard Hamiltonian for the three electronic bands crossing the Fermi level in Sr₂RuO₄, we calculate the band structure and spin susceptibility χ(q, w) in quantitative agreement with nuclear magnetic resonance (NMR) and inelastic neutron scattering (INS) experiments. The susceptibility has two peaks at Qi = (2π/3a, 2π/3a, 0) due to the nesting Fermi surface properties and at qi = (0.2π/a, 0, 0) due to the tendency towards ferromagnetism. Applying spin-fluctuation exchange theory as in layered cuprates we determine from χ(q,w), electronic dispersions, and Fermi surface topology that superconductivity in Sr₂RuO₄ consists of triplet pairing. Using X(q, w) we can exclude s- and d-wave symmetry for the superconducting order parameter. Furthermore, within our analysis and approximations we find that the order parameter will have a node between neighboring RuO₂-planes and that in the RuO₂-plane f_x²-y²-wave and p-wave symmetry are close in energy.