The spectroscopic studies of very neutron-rich nucleus ^{128}Ag have been performed for the first time at the Radioactive Isotope Beam Factory of RIKEN. A new seniority isomer with a half-life of 1.60(7) μs has been identified and is proposed to have a spin-parity of 16^{-} with a maximally aligned configuration comprising three proton holes in the g_{9/2} orbital and one neutron hole in the h_{11/2} orbital. The new level structure in ^{128}Ag is quite well described by shell model calculations without invoking excitations across the Z=50 and N=82 shell gaps, and presents a good case of seniority scheme in odd-odd nuclei in the south vicinity of the double-magic nucleus ^{132}Sn. With a classification of various components of the proton-neutron interaction, the inversion of lowest-lying 9^{-} and 10^{-} states between ^{128}Ag and its neighboring isotone ^{130}In is found to be dynamically ascribed to the seniority-nonconserving proton-neutron interaction components. The structure above 10^{-} up to the 16^{-} isomer in ^{128}Ag shows remarkable similarities to seniority structures in the semimagic nuclei ^{128}Pd and ^{130}Cd. These spectroscopic features in ^{128}Ag indicate that the N=82 shell closure is still robust in silver isotopes.