Cosmological analyses with type Ia supernovae (SNe Ia) often assume a single empirical relation between colour and luminosity (β) and do not account for varying host-galaxy dust properties. However, from studies of dust in large samples of galaxies, it is known that dust attenuation can vary significantly. Here, we take advantage of state-of-the-art modelling of galaxy properties to characterize dust parameters (dust attenuation A_V, and a parameter describing the dust law slope R_V) for 1100 Dark Energy Survey (DES) SN host galaxies. Utilizing optical and infrared data of the hosts alone, we find three key aspects of host dust that impact SN cosmology: (1) there exists a large range (~1-6) of host R_V; (2) high-stellar mass hosts have R_V on average ~0.7 lower than that of low-mass hosts; (3) for a subsample of 81 spectroscopically classified SNe there is a significant (>3σ) correlation between the Hubble diagram residuals of red SNe Ia and the host R_V that when corrected for reduces scatter by $\sim 13{{\ \rm per\ cent}}$ and the significance of the 'mass step' to ~1σ. These represent independent confirmations of recent predictions based on dust that attempted to explain the puzzling 'mass step' and intrinsic scatter (σ_int) in SN Ia analyses....