Electrospinning is a well-established method for the fabrication of polymer biomaterials, including those with core-shell nanofibers. The variability of structures presents a great range of opportunities in tissue engineering and drug delivery by incorporating biologically active molecules such as drugs, proteins, and growth factors and subsequent control of their release into the target microenvironment to achieve therapeutic effect. The object of study is non-woven core-shell PVA–PEG–SiO2@PVA–GO fiber mats assembled by the technology of coaxial electrospinning. The task of the core-shell fiber development was set to regulate the degradation process under external factors. The dual structure was modified with silica nanoparticles and graphene oxide to ensure the fiber integrity and stability. The influence of the nano additives and crosslinking conditions for the composite was investigated as a function of fiber diameter, hydrolysis, and mechanical properties. Tensile mechanical tests and water degradation tests were used to reveal the fracture and dissolution behavior of the fiber mats and bundles. The obtained fibers were visualized by confocal fluorescence microscopy to confirm the continuous core-shell structure and encapsulation feasibility for biologically active components, selectively in the fiber core and shell. The results provide a firm basis to draw the conclusion that electrospun core-shell fiber mats have tremendous potential for biomedical applications as drug carriers, photocatalysts, and wound dressings.