The residuals of the distance moduli of Type Ia supernovae (SNe Ia) relative to a Hubble diagram fit contain information about the inhomogeneity of the Universe, due to weak lensing magnification by foreground matter. By correlating the residuals of the Dark Energy Survey Year 5 SN Ia sample (DES-SN5YR) with extragalactic foregrounds from the DES Y3 Gold catalogue, we detect the presence of lensing at $6.0 \sigma$ significance. This is the first detection with a significance level above $5\sigma$. Constraints on the effective mass-to-light ratios and radial profiles of dark matter haloes surrounding individual galaxies are also obtained. We show that the scatter of SNe Ia around the Hubble diagram is reduced by modifying the standardization of the distance moduli to include an easily calculable de-lensing (i.e. environmental) term. We use the de-lensed distance moduli to recompute cosmological parameters derived from SN Ia, finding in Flat wcold dark matter a difference of $\Delta \Omega _{\rm M} = +0.036$ and $\Delta w = -0.056$ compared to the unmodified distance moduli, a change of $\sim 0.3\sigma$. We argue that our modelling of SN Ia lensing will lower systematics on future surveys with higher statistical power. We use the observed dispersion of lensing in DES-SN5YR to constrain $\sigma _8$, but caution that the fit is sensitive to uncertainties at small scales. Nevertheless, our detection of SN Ia lensing opens a new pathway to study matter inhomogeneity that complements galaxy-galaxy lensing surveys and has unrelated systematics.