The dynamic control of single-photon scattering in a pair of one-dimensional waveguides mediated by a time-modulated atom-cavity system is investigated. Two cases, where the waveguides are coupled symmetrically or asymmetrically to the atom-cavity system, are discussed in detail. The results show that such time-modulated atom-cavity configuration can behave as a dynamical tunable directional single-photon router. The photons with different frequencies can dynamically be routed from the incident waveguide into any ports of the other with a 100% probability via adjusting the modulated amplitude or phases of the time-modulated atom-cavity coupling strengths, associate with the help of the asymmetrical waveguide-cavity couplings. Furthermore, the influence of dissipation on the routing capability is investigated. It is shown that the present single-photon router is robust against the dissipative process of the system, especially the atomic dissipation. These results are expected to be applicable in quantum information processing and design quantum devices with dynamical modulation.