A novel hybrid nanofibrous scaffold prepared with chitosan [containing 1.2 wt% polyethylene oxide (PEO)] and bioactive glass (BG) was fabricated by an electrospinning technique. The morphological and physicochemical properties of scaffolds were studied by scanning electron microscopy (SEM) and spectroscopy. The measurements of tensile strength and water-contact angles suggested that the incorporation of BG into the nanofibers improves the mechanical properties and hydrophilicity of the scaffolds. Biomineralization of the nanofibers was evaluated by soaking them in simulated body fluid (SBF), and the formation of hydroxycarbonate apatite (HCA) layer was determined by EDX and FE-SEM. The results showed that BG-containing nanofibers could induce the formation of HCA on the surface of the composite after 14 days of immersion in SBF. In vitro-cell viability of human mesenchymal stromal cells (hMSCs) on nanofibers was assessed by using the MTT assay. The cell-adhesion results showed that hMSCs were viable at variable time points on the chitosan/PEO/BG nanofiber scaffolds. In addition, the presence of BG enhanced the alkaline phosphatase (ALP) activity of hMSCs cultured on composite scaffolds at day 14 compared to that on pure chitosan/PEO scaffolds. Our results suggest that a chitosan/PEO/BG nanofibrous composite could be a potential candidate for application in tissue engineering.