The first-passage problem of quasi-nonintegrable Hamiltonian systems subject to light linear/nonlinear dampings and weak external/parametric random excitations is investigated here. The motivation is to acquire asymptotic analytical solution of the first-passage rate or the mean first-passage time based on the averaged Itô stochastic differential equation for quasi-nonintegrable Hamiltonian systems. By using the probability current equation and the Laplace integral method, a new method is proposed to obtain the asymptotic analytical expressions for the first-passage rate in the case of high passage threshold. The associated functions such as the reliability function and the probability density function of first-passage time can then be obtained from the first-passage rate. High passage threshold is the crucial condition for the validity of the proposed method. The random bistable oscillator is studied as an illustrative example using the method. The analytical result obtained from the asymptotic analysis shows its consistency with the Kramers formula. A coupled two-degree-of-freedom (2DOF) nonlinear oscillator subjected to stochastic excitations is studied to illustrate the procedure of acquiring the asymptotic analytical solution. The results obtained from the analytical solution agree well with those from numerical simulation, which verifies the accuracy of the proposed method.