We demonstrate that the Berry curvature monopole of nonmagnetic two-dimensional spin-3/2 holes leads to a novel Hall effect linear in an applied in-plane magnetic field B. Remarkably, all scalar and spin-dependent disorder contributions vanish to leading order in B, while there is no Lorentz force and hence no ordinary Hall effect. This purely intrinsic phenomenon, which we term the anomalous planar Hall effect (APHE), provides a direct transport probe of the Berry curvature accessible in all p-type semiconductors. We discuss experimental setups for its measurement.