Epithelial tissues form thin flexible sheets that are subject to external and internal stresses originating from osmotic and mechanical imbalances. How the polarized interfaces of epithelial cells dissipate mechanical energy to avoid fracture is not well understood. We created two different tissue models with opposite polarity, hemicysts and cysts, both enclosing a lumen. We probed resistance to deformation on the level of single cells as well as in-plane dilatation of the entire tissue putting load on cell-cell connections. A viscoelastic model is derived that provides a quantitative understanding of how apical-basal polarization helps to mitigate mechanical stress on short time scales. The basal actomyosin cortex and cell-cell contacts are essential for the elasticity and resilience of tissue, while the apical surface is soft and harbors excess area to alleviate lateral stress, which is important for the tissue to cope with fast changes in Laplace pressure.