In tissue engineering, scaffolds are usually required, which should mimic the native ECM as closely as possible, which means for most tissues a porous and/or fibrous structure. One efficient method for combining the advantageous properties of biopolymers and inorganic substances into a single fibrous structure is electrospinning. In this study, the aim was to integrate the high stiffness of PHB, the flexibility of PCL and the bioactivity of 58S bioactive glass in multifunctional composite (and hybrid) fibrous scaffolds. The PHB/PCL/58S sol−gel bioactive glass scaffolds were fabricated using a combination of electrospinning and the sol-gel method. The fibrous scaffolds were characterized in terms of fiber diameter distribution, mechanical strength, wetting behavior, microstructure and preliminary in vitro cell culture assays. FTIR analysis demonstrated the successful incorporation of the bioactive glass into the polymer blend, which greatly improved the hydrophilicity of PHB/PCL fibermats. The primary biological response of MG-63 osteoblast-like cells in contact with the prepared fibrous scaffolds was evaluated, illustrating that the hybrid scaffold containing the 58S glass sol were not only favorable to MG-63 cell adhesion but also slightly enhanced cell viability and significantly increased ALP activity. In conclusion, it may be said that the scaffolds developed in this study possess most of the main characteristics necessary as a tissue engineering scaffold.