Electrospinning is an advanced manufacturing strategy used to create innovative medical devices from continuous nanoscale fibers that is endowed with tunable biological, chemical, and physical properties. Innovative medical patches manufactured entirely by electrospinning are discussed in this paper, using a specific plant-derived formulation “1 Primary Wound Dressing©” (1-PWD) as an active pharmaceutical ingredient (API). 1-PWD is composed of neem oil (Azadirachta indica A. Juss.) and the oily extracts of Hypericum perforatum (L.) flowers, according to the formulation patented by the ENEA of proven therapeutic efficacy as wound dressings. The goal of this work is to encapsulate this API and demonstrate that its slow release from an engineered electrospun patch can increase the therapeutic efficacy for wound healing. The prototyped patch is a three-layer core–shell membrane, with a core made of fibers from a 1-PWD-PEO blend, enveloped within two external layers made of medical-grade polycaprolactone (PCL), ensuring mechanical strength and integrity during manipulation. The system was characterized via electron microscopy (SEM) and chemical and contact angle tests. The encapsulation, release, and efficacy of the API were confirmed by FTIR and LC-HRMS and were validated via in vitro toxicology and scratch assays.