The modelling of emission spectra of molecules seen in interstellar clouds requires the knowledge of collisional rate coefficients. Among the commonly observed species, N₂H⁺ is of particular interest since it was shown to be a good probe of the physical conditions of cold molecular clouds. Thus, we have calculated hyperfine-structure-resolved excitation rate coefficients of N₂H⁺(X¹Σ⁺) by H₂(j = 0), the most abundant collisional partner in the cold interstellar medium. The calculations are based on a new potential energy surface, obtained from highly correlated ab initio calculations. State-to-state rate coefficients between the first hyperfine levels were calculated, for temperatures ranging from 5 to 70 K. By comparison with previously published N₂H⁺–He rate coefficients, we found significant differences which cannot be reproduced by a simple scaling relationship. As a first application, we also performed radiative transfer calculations, for physical conditions typical of cold molecular clouds. We found that the simulated line intensities significantly increase when using the new H₂ rate coefficients, by comparison with the predictions based on the He rate coefficients. In particular, we revisited the modelling of the N₂H⁺ emission in the LDN 183 core, using the new collisional data, and found that all three of the density, gas kinetic temperature and N₂H⁺ abundance had to be revised.