  Using a three-band Hubbard Hamiltonian we calculate within the
random-phase-approximation the spin susceptibility, $\chi({\bf q},\omega)$, and
NMR spin-lattice relaxation rate, 1/T$_1$, in the normal state of the triplet
superconductor Sr$_2$RuO$_4$ and obtain quantitative agreement with
experimental data. Most importantly, we find that due to spin-orbit coupling
the out-of-plane component of the spin susceptibility $\chi^{zz}$ becomes at
low temperatures two times larger than the in-plane one. As a consequence
strong incommensurate antiferromagnetic fluctuations of the
quasi-one-dimensional $xz$- and $yz$-bands point into the $z$-direction. Our
results provide further evidence for the importance of spin fluctuations for
triplet superconductivity in Sr$_2$RuO$_4$.
