Vacuum Nodes in QCD at $\theta=\pi$: Exact Results

We show that the vacuum functional of 3+1 dimensional non-abelian gauge theories vanishes for some classical field configurations $\psi_0(A)=0$ when the coefficient of the CP violating $\theta$--term becomes $\theta=\pi$ (mod. $2\pi$). Some of these classical configurations are explicitly identified and include sphalerons. The results shed new light into the non-perturbative behavior of non-abelian gauge theories and suggest a relevant role for these classical configurations in the confinement mechanism at $\theta=0$.