This paper focuses on stabilization of second-order oscillatory systems using both feedback control and intentionally introduced time delay. The argument principle is employed as a key technique to divide the parameter space of time delay and controller gain into several regions. Every admissible value in these regions moves the poles of the closed-loop system towards the left of the complex plane such that stability improvement is achieved. The concept of potent asymptotic stability is first introduced, referring to the property that all the poles of the closed-loop system are stable and lie on the left of open-loop system poles in the complex plane. Analytical characterizations on parameter pairs of time delay and controller gain that result in potent asymptotic stability are established. All the poles and corresponding root locus with respect to gain are examined. Numerical examples and simulations are given to illustrate the usefulness and merits of the theoretical results.