This study uses the application of Pinch analysis to optimize an integrated electrolyzer and methanation system, a promising approach for producing green hydrogen and synthetic natural gas (SNG). By leveraging renewable energy sources, such as wind and solar, electrolysis is used to produce hydrogen, which reacts with captured carbon dioxide in the methanation reactor to form methane. This process not only allows for efficient energy storage but also supports the reduction of greenhouse gas emissions. A key focus of this study is the optimization of thermal energy flows within the system, which has not been extensively addressed in the literature. Pinch analysis was applied to identify the critical Pinch point, which revealed the temperature at which the most efficient heat recovery could be achieved. The design of a tailored heat exchanger network led to significant improvements, including a 66.45% reduction in hot utility consumption and an 18.85% reduction in cold utility demand. Overall, the system achieved global energy savings of 31.02%. These results were compared with the existing literature, demonstrating that our approach offers comparable or superior utility savings while addressing challenges, such as the complexity of chemical reactions and system integration. This research highlights the potential for substantial operational cost reductions and increasing sustainability in industrial applications, contributing to the advancement of renewable energy technologies and the decarbonization of energy systems.