  The spin susceptibility of strongly correlated Sr$_2$RuO$_4$ is known to
display a rich structure in reciprocal space, with a prominent peak at ${\bf
Q}_i$=$(0.3,0.3,0)$. It is still debated if the resulting incommensurate
spin-density-wave fluctuations foster unconventional superconductivity at low
temperature or compete therewith. By means of density functional theory
combined with dynamical mean-field theory, we reveal the realistic
multi-orbital signature of the (dynamic) spin susceptibility beyond existing
weak-coupling approaches. The experimental fluctuation spectrum up to 80 meV is
confirmed by theory. Furthermore, the peak at ${\bf Q}_i$ is shown to carry
nearly equal contributions from each of the Ru$(4d)$-$t_\mathrm{2g}$ orbitals,
pointing to a cooperative effect resulting in the dominant spin fluctuations.
