We propose a scheme of fast three-qubit Toffoli quantum gate for ultracold neutral-atom qubits. The scheme is based on Stark-tuned three-body Förster resonances, which we have observed in our recent experiment [D. B. Tretyakov, Phys. Rev. Lett. 119, 173402 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.173402]. The three-body resonance corresponds to a transition when the three interacting atoms change their states simultaneously, and it occurs at a different dc electric field with respect to the two-body Förster resonance. A combined effect of three-body and two-body Förster interactions in external electric and magnetic fields near the three-body resonance results in complex coherent behavior of the populations and phases of collective states of a three-atom system. We have found that it is possible to obtain experimental conditions suitable to implement three-qubit Toffoli gate with 98.3% fidelity and less than 3 μs duration.