We propose and numerically investigate a novel perfect ultra-narrow band absorber based on a metal-dielectric-metal-dielectric-metal periodic structure working at near-infrared region, which consists of a dielectric layer sandwiched by a metallic nanobar array and a thin gold film over a dielectric layer supported by a metallic film. The absorption efficiency and ultra-narrow band of the absorber are about 98 % and 0.5 nm, respectively. The high absorption is contributed to localized surface plasmon resonance, which can be influenced by the structure parameters and the refractive index of dielectric layer. Importantly, the ultra-narrow band absorber shows an excellent sensing performance with a high sensitivity of 2400 nm/RIU and an ultra-high figure of merit of 4800. The FOM of refractive index sensor is significantly improved, compared with any previously reported plasmonic sensor. The influences of structure parameters on the sensing performance are also investigated, which will have a great guiding role to design high-performance refractive index sensors. The designed structure has huge potential in sensing application.