We present a MoS2–graphene bilayer-based high-performance refractive index surface plasmon resonance (SPR) sensor applying an angular interrogation technique. We used it for inspecting the reflected optical signal from the sensing medium. For enhancing the sensor angular sensitivity (S), signal-to-noise ratio (SNR), and quality factor (QF), we undertook some steps sequentially. First, the impact of gold layer thickness was investigated and optimized to 40 nm. Second, the MoS2 and graphene coating layers were optimized to four and three, respectively. Finally, the minimum reflectance and SPR angle were identified with this optimum structure. It is seen that the angular sensitivity of this optimum structure improved to an excellent value of 130  deg-RIU^−  1 along with an improved angular SNR of 1.37 and QF of 17.02  RIU^−  1. Then we compared the proposed sensor with the existing works in terms of angular sensitivity, SNR, and QF to prove its effective application. Finally, an analysis of DNA hybridization is presented as an application of the proposed structure. Our offered configuration is capable of recognizing absorption of DNAs by means of the attenuated total reflection method. The effective refractive index of the sensing medium alters because of the absorption of various concentrated DNAs. Computational investigation demonstrates that the change of SPR angle and minimum reflectance for mismatched DNA strands is inconsiderable, though it is significant for complementary DNA strands, which is crucial for the sensing of accurate DNA hybridization.