This paper reports a rational design of temperature-responsive nanofiber meshes with shape-memory effect. The meshes were fabricated by electrospinning a poly(ε-caprolactone) (PCL)-based polyurethane with different contents of soft and hard segments. The effects of PCL diol/hexamethylene diisocyanate (HDI)/1,4-butanediol (BD) molar ratio in terms of the contents of soft and hard segments on the shape-memory properties were investigated. Although the mechanical property improved with increasing hard segment ratio, optimal shape-memory properties were obtained with a PCL/HDI/BD molar ratio of 1:4:3. At a microscopic level, the original nanofibrous structure was easily deformed into a temporary shape, and recovered its original structure when the sample was reheated. A higher recovery rate (>89%) was achieved even when the mesh was deformed up to 400%. Finally, the nanofiber meshes were used to control the alignment of human mesenchymal stem cells (hMSCs). The hMSCs aligned well along the fiber orientation. The proposed nanofibrous meshes with the shape-memory effect have the potential to serve as in vitro platforms for the investigation of cell functions as well as implantable scaffolds for wound-healing applications.