Spacecraft magnetometry and paleomagnetic measurements of lunar samples provide evidence that the Moon had a magnetic field billions of years ago. Because this field was likely stronger than that predicted by scaling laws for core convection dynamos, a longstanding hypothesis is that an ancient dynamo was amplified by plasma from basin-forming impacts. However, there have been no self-consistent models that quantify whether this process can generate the required field intensities. Our impact and magnetohydrodynamic simulations show that for an initial maximum surface field of only 2 microtesla, plasmas created from basin-forming impacts can amplify a planetary dipole field at the basin antipode to ~43 microtesla. This process, coupled with impact-induced body pressure waves focusing at the antipode, could produce magnetization that can account for the crustal fields observed today.