In this work, a sensitive sensing platform was developed using a glassy carbon electrode (GCE) to simultaneously determine AA, DA, and UA. Cu nanostructures, the poly-L-Lysine (p(L-Lys)), and the electrochemically reduced graphene oxide (ERGO) modified GCE (GCE/Cu@ERGO-p(L-Lys)) was developed via the electrodeposition of Cu and electropolymerization of the ERGO-p(L-Lys). Simultaneous detection makes analysis more efficient and cost-effective by reducing the need for multiple sensors. The GCE/Cu@ERGO-p(L-Lys) was characterized by electrochemical impedance spectroscopy, cyclic voltammetry, and scanning electron microscopy. The limit of detection values for AA, DA, and UA analytes were 0.16, 0.033, and 0.021 μM, respectively, while the linear ranges were 0.53–50.0, 0.11–100.0, and 0.070–0.75 μM. The proposed sensor was found to be applicable for the determination of target analytes in fetal bovine serum samples. The proposed GCE/Cu@ERGO-p(L-Lys) hybrid composite modified electrode is a promising material for simultaneous determination in biological fluids with excellent analytical performance and anti-interference effect.