In this study, for the first time, core/shell nanofibrous mats of polycaprolactone (PCL) and chitosan (CS) as a carrier for the rosuvastatin (RSV) drug were fabricated. To do this, the electrospinning technique using a two-fluid stainless steel coaxial spinneret consisted of two concentrically arranged needles was used. The CS layer as a shell contained 5% (w/w) of the RSV drug. The fabricated nanofibrous mats were characterized by different techniques, including scanning electron microscopy (SEM), atomic force microscopy (AFM), transmission electron microscopy (TEM), Fourier transform infrared spectroscopy (FTIR), tensile testing, and hydrophilicity measurements. Then, the fabricated core/shell nanofibrous mats containing the RSV drug as a drug delivery system (DDS) were used to evaluate the in vitro drug release behavior in a typical phosphate-buffered saline (PBS) solution at different pH values (i.e., 4, 6, and 7.4) for 48 h. Moreover, the corresponding drug release mechanism was investigated. To study the cytotoxicity, the viability of the human fibroblast cells exposed to the fabricated DDS was examined. The results showed that under optimum electrospinning conditions, the average diameter of the PCL core was about 120 nm, and the thickness of the CS shell was about 60 nm. Also, the presence of all the components was confirmed in the structure of the fabricated DDS. Moreover, the addition of the RSV drug had no significant effect on the mechanical properties of the PCL/CS core/shell nanofibrous mats. Due to the pH-responsive feature of the CS shell, the RSV drug release from the fabricated DDS showed a reasonable environmental response as the pH value of the PBS solution decreased, the degree of drug release correspondingly increased. It was also found that the prolonged release of the RSV drug from the fabricated DDS followed the Korsmeyer–Peppas kinetic model with the Fickian diffusion mechanism at all the pH conditions. The CS layer enhanced the cytotoxicity and hydrophilicity of the fabricated DDS and led to the controllable drug release behavior, which would provide a beneficial approach for drug delivery technology.