An implicit direct-time integration method for obtaining transient responses of general dynamic systems is described. The conventional Newmark method cannot be directly applied to state-space first-order differential equations, which contain no explicit acceleration terms. The method proposed here is the statespace Newmark method that incorporates the average velocity concept, and can be applied to an analysis of general dynamic systems that are expressed by state-space first-order differential equations. It is also readily coded into a program. Stability and accuracy analyses indicate that the method is numerically unconditionally stable like the conventional Newmark method, and has a period error of second-order accuracy for small damping and fourth-order for large damping and an amplitude error of second-order, regardless of damping. In addition, its utility and validity are confirmed by two application examples. The results suggest that the proposed state-space Newmark method based on average velocity be generally applied to the analysis of transient responses of general dynamic systems with a high degree of reliability with respect to stability and accuracy.