Asphalt concrete (AC) exhibits significant tension-compression (TC) asymmetry, which is currently not considered in pavement design. This study develops a novel temperature-dependent dual viscoelastic model to quantitatively capture the viscoelastic behaviour of AC. Unlike the conventional viscoelastic constitutive model, the proposed model decomposes strain into tensile and compressive components to characterise AC’s TC asymmetry. Additionally, a systematic modelling framework with intrinsic TC asymmetry is developed for the first time to predict the response of pavement under moving tire load. The results illustrate that implementing the proposed dual viscoelastic model enlarges both the vertical deformation of pavements and the tensile and shear strains in the AC layers, bringing it closer to the realistic scenario compared to the conventional model that only considers compression properties. Furthermore, high temperatures and low vehicular speeds exacerbate the substantial effects of AC’s TC asymmetry on asphalt pavement. This study provides a valuable method to capture AC’s TC asymmetry and predict pavement response more accurately, giving better insight into pavement response and enhancing pavement design and maintenance.