We present a new field-theoretic approach to analyze nonequilibrium dynamics of two-level systems (TLS), which is based on a correspondence between a driven TLS and a Majorana fermion field theory coupled to bosonic fields. This approach allows us to calculate analytically properties of nonlinear TLS dynamics with an arbitrary accuracy. We apply our method to analyze specific TLS dynamics under a monochromatic periodic drive that is relevant to the problem of decoherence in Josephson junction qubits. It is demonstrated that the method gives the precise positions of the resonance peaks in the nonlinear dielectric response function that are in agreement with numerical simulations.