Alpine catchments show a high sensitivity to climate variation as they include the elevation range of the snow line. Therefore, the correct representation of climate variables and their interdependence is crucial when describing or predicting hydrological processes. When using climate model simulations in hydrological impact studies, forcing meteorological data are usually downscaled and bias corrected, most often by univariate approaches such as quantile mapping of individual variables, neglecting the relationships that exist between climate variables. In this study we test the hypothesis that the explicit consideration of the relation between air temperature and precipitation will affect hydrological impact modelling in a snow-dominated mountain environment. Glacio-hydrological simulations were performed for two partly glacierized alpine catchments using a recently developed multivariate bias correction method to post-process EURO-CORDEX regional climate model outputs between 1976 and 2099. These simulations were compared to those obtained by using the common univariate quantile mapping for bias correction. As both methods correct each climate variable's distribution in the same way, the marginal distributions of the individual variables show no differences. Yet, regarding the interdependence of precipitation and air temperature, clear differences are notable in the studied catchments. Simultaneous correction based on the multivariate approach led to more precipitation below air temperatures of 0 ∘C and therefore more simulated snowfall than with the data of the univariate approach. This difference translated to considerable consequences for the hydrological responses of the catchments. The multivariate bias-correction-forced simulations showed distinctly different results for projected snow cover characteristics, snowmelt-driven streamflow components, and expected glacier disappearance dates. In all aspects – the fraction of precipitation above and below 0 ∘C, the simulated snow water equivalents, glacier volumes, and the streamflow regime – simulations resulting from the multivariate-corrected data corresponded better with reference data than the results of univariate bias correction. Differences in simulated total streamflow due to the different bias correction approaches may be considered negligible given the generally large spread of the projections, but systematic differences in the seasonally delayed streamflow components from snowmelt in particular will matter from a planning perspective. While this study does not allow conclusive evidence that multivariate bias correction approaches are generally preferable, it clearly demonstrates that incorporating or ignoring inter-variable relationships between air temperature and precipitation data can impact the conclusions drawn in hydrological climate change impact studies in snow-dominated environments.