Wound healing is a complex tissue regeneration process that promotes the growth of new tissue to provide the body with the necessary barrier from the outside environment. In the class of non-healing wounds, diabetic wounds, and ulcers, dressing materials that are available clinically (e.g., gels and creams) have demonstrated only a slow improvement with current available technologies. Among all available current technologies, electrospun fibers exhibit several characteristics that may provide novel replacement dressing materials for the above-mentioned wounds. Therefore, in this review, we focus on recent achievements in electrospun drug-eluting fibers for wound healing applications. In particular, we review drug release, including small molecule drugs, proteins and peptides, and gene vectors from electrospun fibers with respect to wound healing. Furthermore, we provide an overview on multifunctional dressing materials based on electrospun fibers, including those that are capable of achieving wound debridement and wound healing simultaneously as well as multi-drugs loading/types suitable for various stages of the healing process. Our review provides important and sufficient information to inform the field in development of fiber-based dressing materials for clinical treatment of non-healing wounds.