Recently, substantial interest has been generated in using electrospun biomimetic nanofibers of hybrids, particularly organic/inorganic, to engineer different tissues. The present work, for the first time, introduced a unique natural and synthetic hybrid micronanofiber wound dressing, composed of virgin olive oil/copper oxide nanocrystals and polyurethane (PU), developed via facile electrospinning. The as-spun organic/inorganic hybrid micronanofibers were characterized by scanning electron microscopy (SEM), energy dispersive X-ray analysis, X-ray diffraction, electron probe microanalysis, and transmission electron microscopy. The interaction of cells with scaffold was studied by culturing NIH 3T3 fibroblasts on an as-spun hybrid micronanofibrous mat, and viability, proliferation, and growth were assessed. The 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide assay results and SEM observation showed that the hybrid micronanofibrous scaffold was noncytotoxic to fibroblast cell culture and was found to benefit cell attachment and proliferation. Hence our results suggest the potential utilization of as-spun micronanoscaffolds for tissue engineering. Copper oxide–olive oil/PU wound dressing may exert its positive beneficial effects at every stage during wound-healing progression, and these micronanofibers may serve diverse biomedical applications, such as tissue regeneration, damaged skin treatment, wound healing applications, etc. Conclusively, the fabricated olive oil–copper oxide/PU micronanofibers combine the benefits of virgin olive oil and copper oxide, and therefore hold great promise for biomedical applications in the near future.