The extinction spectra and electric field distribution of an asymmetric cylindrical nanorod dimer (ACND) are calculated by discrete dipole approximation. The ACND is composed of two linear orders of cylindrical silver nanorods with different radii and lengths. The effects of the structural parameters of ACND on the localized surface plasmon resonance (LSPR) mode are also studied. Results show two resonance peaks in the extinction spectra of ACND: the higher-energy anti-bonding mode and the lower-energy bonding mode. The interaction of two hybridization plasmonic resonance modes produces an asymmetric line shape in the extinction spectra, which is considered to be a Fano resonance profile.