Plasmonic biosensing has been demonstrated to be a powerful technique for quantitative determination of molecular analytes and kinetic analysis of biochemical reactions. However, interfaces of most plasmonic biosensors are made of noble metals, such as gold and silver, which are not compatible with industrial production technologies. This greatly limits biosensing applications beyond biochemical and pharmaceutical research. Here, we propose and investigate copper-based biosensor chips fully fabricated with a standard complementary metal-oxide-semiconductor (CMOS) process. The protection of thin copper films from oxidation is achieved with SiO2 and Al2O3 dielectric films deposited onto the metal surface. In addition, the deposition of dielectric films with thicknesses of only several tens of nanometers significantly improves the biosensing sensitivity, owing to better localization of electromagnetic field above the biosensing surface. According to surface plasmon resonance (SPR) measurements, the copper biosensor chips coated with thin films of SiO2 (25 nm) and Al2O3 (15 nm) show 55% and 75% higher sensitivity to refractive index changes, respectively, in comparison to pure gold sensor chips. To test biomolecule immobilization, the copper-dielectric biosensor chips are coated with graphene oxide linking layers and used for the selective analysis of oligonucleotide hybridization. The proposed plasmonic biosensors make SPR technology more affordable for various applications and provide the basis for compact biosensors integrated with modern electronic devices.