The XENON1T experiment searches for dark matter particles through their scattering off xenon atoms in a 2 tonne liquid xenon target. The detector is a dual-phase time projection chamber, which measures simultaneously the scintillation and ionization signals produced by interactions in target volume, to reconstruct energy and position, as well as the type of the interaction. The background rate in the central volume of XENON1T detector is the lowest achieved so far with a liquid xenon-based direct detection experiment. In this work we describe the response model of the detector, the background and signal models, and the statistical inference procedures used in the dark matter searches with a 1 tonne$\times$year exposure of XENON1T data, that leaded to the best limit to date on WIMP-nucleon spin-independent elastic scatter cross-section for WIMP masses above 6 GeV/c$^2$.