In this study, we investigate the stability and asymptotic stability properties of Caputo fractional time-dependent systems with delay by employing vector Lyapunov functions. Utilizing the Caputo fractional Dini derivative on Lyapunov-like functions, along with a new comparison theorem and differential inequalities, we derive and prove sufficient conditions for the stability and asymptotic stability of these complex systems. An example is included to showcase the method's practicality and to specifically illustrate its advantages over scalar Lyapunov functions. Our results improves, extends, and generalizes several existing findings in the literature.