The reliability analysis of energy-dissipation bridge structures with viscous dampers under multi-support seismic excitations is a typical local nonlinear dynamic problem. The Monte Carlo simulation method has the general applicability to solve the nonlinear dynamic reliability problem，but with the problem of huge computational cost for engineering practice. The computational cost is determined by the number of samples and the efficiency of a single run of deterministic structural dynamic analysis. In order to improve the efficiency of the seismic reliability analysis of the energy-dissipation bridge structures with viscous dampers，an explicit time-domain dimension-reduced iteration scheme is established using precise time-integration method and Newton-Raphson method，so that the efficiency of dynamic analysis is improved. A subset simulation method using Hamiltonian Monte Carlo is introduced to improve the sampling efficiency of random ground motion in failure domain. Numerical results show that the high efficiency of the present approach for solving nonlinear dynamic reliability problems of energy-dissipation bridge structures with viscous dampers.