Low-energy incommensurate (IC) magnetic fluctuations in the multiband superconductor Sr$_2$RuO$_4$ is investigated by high-resolution inelastic neutron scattering measurements and random phase approximation (RPA) calculations. Below $T_text{c}$, the substantial spin gap is observed at $mathbf{Q}_text{IC}=(0.3, 0.3, L)$ where the quasi-one-dimensional $alpha$ and $beta$ sheets consisting of the Fermi surfaces are in good nesting conditions. $L$ modulated intensity of the low-energy IC magnetic fluctuations and our RPA calculations indicate that the superconducting gaps regarding the $alpha$ and $beta$ sheets have the horizontal line nodes.