This paper considers the eigenvalue distribution of a linear time-invariant (LTI) system with commensurate time delays and its application to proportional-integral-derivative (PID) controller design for a delay plant via dominant eigenvalue assignment. A new result on the root distribution of a quasi-polynomial is first produced by applying part of Pontryagin's conclusions. This result which gives a necessary and sufficient condition can be directly used to judge the number of the right-half plane eigenvalues of the characteristic equation of a time delay system. Based on the proposed result, necessary and sufficient conditions on dominant eigenvalue assignment for PID control of time delay systems are presented and an algorithm is then provided to determine the PID controller gains. The proposed approaches can assign some (one or two) eigenvalues to the desired positions and all the other eigenvalues to the left of a given line to guarantee the dominance of the assigned ones, which enables us to design the controller according to the desired performance indexes for a standard first-order system or a standard second-order system in addition to stability. The method is effective for the closed-loop characteristic equation being retarded type or neutral type. The controller gains to achieve the control objective can be characterized by a straightforward computation. Further, a result on degradation to proportional-integral (PI) control of time delay systems is given.