We present Perov's type $(\beta ,F)$-contraction principle and examine the fixed points of the self-operators satisfying Perov's type $(\beta ,F)$-contraction principle in the context of vector-valued $b$-metrics. A specific instance of the $(\beta ,F)$-contraction principle is the $F$-contraction principle. We generalize a number of recent findings that are already in the literature and provide an example to illustrate the hypothesis of the main theorem. We apply the obtained fixed point theorem to show the existence of the solution to the delay integro-differential problem.