The cells and tissue in the human body are orderly and directionally arranged, and constructing an ideal biomimetic extracellular matrix is still a major problem to be solved in tissue engineering. In the field of the bioresorbable vascular grafts, the long-term functional prognosis requires that cells first migrate and grow along the physiological arrangement direction of the vessel itself. Moreover, the graft is required to promote the formation of neointima and the development of the vessel walls while ensuring that the whole repair process does not form a thrombus. In this study, poly (l-lactide-co-ε-caprolactone) (PLCL) shell layers and polyethylene oxide (PEO) core layers with different microstructures and loaded with sodium tanshinone IIA sulfonate (STS) were prepared by coaxial electrospinning. The mechanical properties proved that the fiber membranes had good mechanical support, higher than that of the human aorta, as well as great suture retention strengths. The hydrophilicity of the oriented-fiber membranes was greatly improved compared with that of the random-fiber membranes. Furthermore, we investigated the biocompatibility and hemocompatibility of different functional fiber membranes, and the results showed that the oriented-fiber membranes containing sodium tanshinone IIA sulfonate had an excellent antiplatelet adhesion effect compared to other fiber membranes. Cytological analysis confirmed that the functional fiber membranes were non-cytotoxic and had significant cell proliferation capacities. The oriented-fiber membranes induced cell growth along the orientation direction. Degradation tests showed that the pH variation range had little change, the material mass was gradually reduced, and the fiber morphology was slowly destroyed. Thus, results indicated the degradation rate of the oriented-fiber graft likely is suitable for the process of new tissue regeneration, while the random-fiber graft with a low degradation rate may cause the material to reside in the tissue for too long, which would impede new tissue reconstitution. In summary, the oriented-functional-fiber membranes possessing core-shell structures with sodium tanshinone IIA sulfonate/polyethylene oxide loading could be used as tissue engineering materials for applications such as vascular grafts with good prospects, and their clinical application potential will be further explored in future research.