Detection of weak electromagnetic waves and hypothetical particles aided by quantum amplification is important for fundamental physics and applications. However, demonstrations of quantum amplification are still limited; in particular, the physics of quantum amplification is not fully explored in periodically driven (Floquet) systems, which are generally defined by time-periodic Hamiltonians and enable observation of many exotic quantum phenomena such as time crystals. Here we investigate the magnetic-field signal amplification by periodically driven ^{129}Xe spins and observe signal amplification at frequencies of transitions between Floquet spin states. This "Floquet amplification" allows us to simultaneously enhance and measure multiple magnetic fields with at least one order of magnitude improvement, offering the capability of femtotesla-level measurements. Our findings extend the physics of quantum amplification to Floquet spin systems and can be generalized to a wide variety of existing amplifiers, enabling a previously unexplored class of "Floquet spin amplifiers".