Tailings dam-breach analyses are essential for flood-hazard assessment, emergency planning and risk estimation, but their results are strongly affected by uncertainties in breach development, released volume and tailings rheology. This study proposes an efficient probabilistic methodology that integrates uncertainty quantification and global sensitivity analysis for tailings dam-breach studies. High-dimensional outputs (spatial maps) and the computational cost of deterministic simulations are addressed through dimensionality reduction and metamodeling. The methodology is demonstrated on a benchmark case with complex terrain using HEC-RAS v6.6 and considering uncertainties in breach parameters and rheological properties. The results quantify uncertainty in maximum flow depth and arrival time, characterize their statistical distributions and identify the spatial influence of the main input variables through sensitivity maps. Sensitivity indices reveal the dominance of breach parameters near the dam and yield stress farther downstream. The modular and non-intrusive framework can be coupled with other deterministic models and applied to different dam-breach scenarios, supporting more standardized and risk-informed assessments.