Muons from cosmic-ray interactions in the atmosphere provide a high-statistics source of particles that can be used to study the performance and calibration of the ATLAS detector. Cosmic-ray muons can penetrate to the cavern and deposit energy in all detector subsystems. Such events have played an important role in the commissioning of the detector since the start of the installation phase in 2005 and were particularly important for understanding the detector performance in the time prior to the arrival of the first LHC beams. Global cosmic-ray runs were undertaken in both 2008 and 2009 and these data have been used through to the early phases of collision data-taking as a tool for calibration, alignment and detector monitoring. These large datasets have also been used for detector performance studies, including investigations that rely on the combined performance of different subsystems. This paper presents the results of performance studies related to combined tracking, lepton identification and the reconstruction of jets and missing transverse energy. Results are compared to expectations based on a cosmic-ray event generator and a full simulation of the detector response.