This paper introduces a novel plasmon refractive index nanosensor structure based on Fano resonance. The structure comprises a metal–insulator–metal (MIM) waveguide with an inverted rectangular cavity and a circle minus a small internal circle plus a rectangular cavity (CMSICPRC). This study employs the finite element method (FEM) to analyze the sensing characteristics of the structure. The results demonstrate that the geometrical parameters of specific structures exert a considerable influence on the sensing characteristics. Simulated experimental data show that the maximum sensitivity of this structure is 3240 nm/RIU, with a figure of merit (FOM) of 52.25. Additionally, the sensor can be used in biology, for example, to detect the concentration of hemoglobin in blood. The sensitivity of the sensor in this application, according to our calculations, can be 0.82 nm∙g/L.