We propose a plasmonic perfect absorber (PPA) based on metal nanorod arrays with the connected veins for ultra-sensitive refractive index (RI) sensing applications in the near-infrared region. The physical origin is explained through the absorptance/reflectance spectra, distribution of electric/magnetic field intensity, surface charge density, height and gap distance. The designed PPA can constitute a cavity resonance center and serve as a plasmonic sensor. There is a huge difference of absorptance and reflectance between the case with and without veins due to the vein effect has a dominant influence on plasmon resonance in the proposed PPA. Simulation results show that the absorptance (A) of the proposed PPA is nearly perfect (A = 99.67%) which is about 1332.49 times enhancement compared to its counterpart without the connected veins (A = 0.075%). The calculated sensitivity, figure of merit and quality factor can be achieved up to 800.00 nm/RIU (RIU is the refractive index unit), 12.17 (RIU−1) and 14.78, respectively.