A simultaneous measurement of three Standard Model cross-sections using 4.7 inverse femtobarns of proton-proton collision data at a center-of-mass energy of 7 TeV is presented. Collision data were collected using the ATLAS detector at the Large Hadron Collider. The signal production cross-sections studied are for top quark pair production, charged weak boson pair production, and Drell-Yan production of tau lepton pairs with invariant mass greater than 40 GeV. A data sample is defined from events with isolated high-energy electron-muon pairs arranged in a phase space defined by missing transverse momentum and jet multiplicity. A binned maximum likelihood fit is employed to determine signal yields in this phase space. Signal event yields are in turn used to measure full cross-section values and cross-section values within a fiducial region of the detector, and unlike conventional measurements the signal measurements are performed simultaneously. This is the first such simultaneous measurement of these cross-sections using the ATLAS detector. Measured cross-sections are found in good agreement with the most precise published theoretical predictions.