The tracking performance parameters of the ATLAS Transition Radiation Tracker (TRT) as part of the ATLAS inner detector are described in this paper for different data-taking conditions in proton-proton, proton-lead and lead-lead collisions at the Large Hadron Collider (LHC). The performance is studied using data collected during the first period of LHC operation (Run 1) and is compared with Monte Carlo simulations. The performance of the TRT, operating with two different gas mixtures (xenon-based and argon-based) and its dependence on the TRT occupancy is presented. These studies show that the tracking performance of the TRT is similar for the two gas mixtures and that a significant contribution to the particle momentum resolution is made by the TRT up to high particle densities.