We present cosmological constraints from a joint cosmic shear analysis of the Kilo-Degree Survey (KV450) and the Dark Energy Survey (DES-Y1), which were conducted using Complete Orthogonal Sets of E/B-Integrals (COSEBIs). With COSEBIs, we isolated any B-modes that have a non-cosmic shear origin and demonstrate the robustness of our cosmological E-mode analysis as no significant B-modes were detected. We highlight how COSEBIs are fairly insensitive to the amplitude of the non-linear matter power spectrum at high k-scales, mitigating the uncertain impact of baryon feedback in our analysis. COSEBIs, therefore, allowed us to utilise additional small-scale information, improving the DES-Y1 joint constraints on S₈ = σ₈(Ω_m/0.3)^0.5 and Ω_m by 20%. By adopting a flat ΛCDM model we find S₈ = 0.755_−0.021^+0.019, which is in 3.2σ tension with the Planck Legacy analysis of the cosmic microwave background.