We explore how to apply perturbation theory to complicated time-dependent Hamiltonian systems that involve complex potentials. To do this, we introduce a generalized time-dependent oscillator to which the complex potentials are connected through a weak coupling strength. We regard the complex potentials in the Hamiltonian as the perturbed terms. Quantum characteristics of the system, such as wave functions and expectation values of the Hamiltonian, are investigated on the basis of the perturbation theory. We apply our theory to particular systems with explicit choices of time-dependent parameters. Through such applications, the time behavior of the quantum wave packets and the spectrum of expectation values of the Hamiltonian are analyzed in detail. We confirm that the imaginary parts of expectation values of the Hamiltonian are not zero but very small, whereas the real parts deviate slightly from those of the unperturbed system.