This research study reports the development of chitosan/carboxylated graphene oxide (CS/GO-COOH) composite scaffolds with nanofibrous architecture using the electrospinning method. The concept of designed composite fibrous material is based on bringing together the biological properties of CS, mechanical, electrical, and biological characteristics of GO-COOH with the versatility and efficiency of ultra-modern electrospinning techniques. Three different concentrations of GO-COOH were added into a chitosan (CS)-poly(ethylene oxide) (PEO) solution (the ratio between CS/PEO was 3/7 (w/w)) and were used in the synthesis process of composite scaffolds. The effect of GO-COOH concentration on the spinnability, morphological and mechanical features, wettability, and biological properties of engineered fibrous scaffolds was thoroughly investigated. FTIR results revealed the non-covalent and covalent interactions that could take place between the system’s components. The SEM micrographs highlighted the nanofibrous architecture of scaffolds, and the presence of GO-COOH sheets along the composite CS/GO-COOH nanofibers. The size distribution graphs showed a decreasing trend in the mean diameter of composite nanofibers with the increase in GO-COOH content, from 141.40 nm for CS/PG 0.1% to 119.88 nm for CS/PG 0.5%. The dispersion of GO-COOH led to composite scaffolds with increased elasticity; the Young’s modulus of CS/PG 0.5% (84 ± 4.71 MPa) was 7.5-fold lower as compared to CS/PEO (662 ± 15.18 MPa, p < 0.0001). Contact angle measurements showed that both GO-COOH content and crosslinking step influenced the surface wettability of scaffolds, leading to materials with ~1.25-fold higher hydrophobicity. The in vitro cytocompatibility assessment showed that the designed nanofibrous scaffolds showed a reasonable cellular proliferation level after 72 h of contact with the fibroblast cells.