We investigate cross sections for neutrino-12C exclusive scattering and for muon capture on 12C using wave functions obtained in the ab initio no-core shell model. In our parameter-free calculations with basis spaces up to the 6 variant Planck's over 2pi Omega we show that realistic nucleon-nucleon interactions, like, e.g., the CD-Bonn, underpredict the experimental cross sections by more than a factor of 2. By including a realistic three-body interaction, Tucson-Melbourne TM'(99), the cross sections are enhanced significantly and a much better agreement with experiment is achieved. At the same time, the TM'(99) interaction improves the calculated level ordering in 12C. The comparison between the CD-Bonn and the three-body calculations provides strong confirmation for the need to include a realistic three-body interaction to account for the spin-orbit strength in p-shell nuclei.