The development of nanofibrous membranes with tunable wettability, degradation, and biocompatibility is highly keen for biomedical applications, including drug delivery and wound dressing. In this study, biocompatible and biodegradable nanofibrous membranes with antioxidant properties were successfully prepared by the electrospinning technique. The membranes were developed using polylactic acid (PLA) and polyethylene oxide (PEO) as the matrix, with the addition of grape seed extract (GSE), a rich source of natural antioxidants. The nanofibrous membranes were thoroughly characterized both from the materials and from the biocompatibility point of view. PLA and PLA/PEO nanofibers showed high encapsulation efficiency, close to 90%, while the encapsulated GSE retained its antioxidant capacity in the membranes. In vitro release studies showed that GSE diffuses from PLA/GSE and PLA/PEO/GSE membranes in a Fickian diffusion manner, whose experimental data were well fitted using the Korsmeyer-Peppas model. Furthermore, a higher controlled release of GSE was observed for the PLA/PEO/GSE membrane. Moreover, culturing experiments with human foreskin fibroblast (HFF1) cells demonstrated that all samples are biocompatible and showed that the GSE-loaded PLA/PEO nanofibrous membranes support better cell attachment and proliferation compared to the PLA/GSE nanofibrous membranes, owing to the superior hydrophilicity. In summary, the results suggested that the GSE-loaded membranes are a promising topical drug delivery system and have a great potential for wound dressing applications.