We present cosmological constraints from Dark Energy Survey Year 3 (DES Y3) weak lensing data using persistent homology, a topological data analysis technique that tracks how features like clusters and voids evolve across density thresholds. For the first time, we apply spherical persistent homology to galaxy survey data through the algorithm TOPOS2, which is optimized for curved-sky analyses and HEALPIX compatibility. Employing a simulation-based inference framework with the Gower Street simulation suite specifically designed to mimic DES Y3 data properties we extract topological summary statistics from convergence maps across multiple smoothing scales and redshift bins. After neural network compression of these statistics, we estimate the likelihood function and validate our analysis against baryonic feedback effects, finding minimal biases (under) in the $0.3\sigma$ plane. Assuming the wCold Dark Matter model, our combined Betti numbers and second moments analysis yields $\Omega _\mathrm{m}-S_8$ and $S_8 = 0.821 \pm 0.018$ constraints 70 per cent tighter than those from cosmic shear two-point statistics in the same parameter plane. Our results demonstrate that topological methods provide a powerful and robust framework for extracting cosmological information, with our spherical methodology readily applicable to upcoming Stage IV wide-field galaxy surveys. $\Omega _\mathrm{m} = 0.304\pm 0.037$
