Cardiovascular disease is anticipated to remain the leading cause of death globally. Due to the current problems connected with using autologous arteries for bypass surgery, researchers are developing tissue-engineered vascular grafts (TEVGs). The major goal of vascular tissue engineering is to construct prostheses that closely resemble native blood vessels in terms of morphological, mechanical, and biological features so that these scaffolds can satisfy the functional requirements of the native tissue. In this setting, morphology and cellular investigation are usually prioritized, while mechanical qualities are generally addressed superficially. However, producing grafts with good mechanical properties similar to native vessels is crucial for enhancing the clinical performance of vascular grafts, exposing physiological forces, and preventing graft failure caused by intimal hyperplasia, thrombosis, aneurysm, blood leakage, and occlusion. The scaffold’s design and composition play a significant role in determining its mechanical characteristics, including suturability, compliance, tensile strength, burst pressure, and blood permeability. Electrospun prostheses offer various models that can be customized to resemble the extracellular matrix. This review aims to provide a comprehensive and comparative review of recent studies on the mechanical properties of fibrous vascular grafts, emphasizing the influence of structural parameters on mechanical behavior. Additionally, this review provides an overview of permeability and cell growth in electrospun membranes for vascular grafts. This work intends to shed light on the design parameters required to maintain the mechanical stability of vascular grafts placed in the body to produce a temporary backbone and to be biodegraded when necessary, allowing an autologous vessel to take its place.