For the first time, this study reports the development and investigation of dual-function mixed matrix membranes (MMMs) based on poly(ether-block-amide) (PEBA) modified with the copper-based metal–organic framework HKUST-1, possessing advanced properties and designed for pervaporation (PV) and gas separation (GS). A systematic study of HKUST-1 incorporation (5–15 wt%) into PEBA revealed that optimal 10 wt% loading improved permeation flux and selectivity by 1.9- and 1.1-fold, respectively, for isopropanol/water (5/95 wt%) separation. The supported membrane showed a 10.5-fold increase in flux over dense PEBA. Furthermore, PV of isopropanol/phenol aqueous mixtures at elevated temperatures and GS tests confirmed sustained enhanced transport properties with higher permeability. Specifically, the CO2 and CH4 permeabilities increased by 24.4% and 165.3%, respectively, compared to the unmodified supported membrane. To elucidate the structure–property relationships, various techniques (Fourier-transform infrared spectroscopy, thermogravimetric analysis, scanning electron and atomic force microscopies, water contact angle, and swelling degree measurements) were employed. DFT calculations were performed to investigate non-covalent interactions in hypothetical molecular associates. The demonstrated efficiency in both PV and GS underscores the versatility of PEBA/HKUST-1 MMMs, positioning them as promising candidates for advanced membrane processes.