Copper based catalysts have high potential for the substituent of noble-metal based catalysts as their high selectivity and moderate activity for selective hydrogenation reaction; however, achieving further high catalytic stability is very difficult. In this work, the carbonization process of Cu-based organic frameworks was explored for the synthesis of highly-dispersed Cu supported by hierarchically porous carbon with high catalytic performance for selective hydrogenation of 1,3-butadiene. The porous hierarchy of carbon support and the dispersion of copper nanoparticles can be precisely tuned by controlling the carbonization process. The resultant catalyst carbonized at 600 °C exhibits a rather low reaction temperature at 75 °C for 100% butadiene conversion with 100% selectivity to butenes, due to its reasonable porous hierarchy and highly-dispersed copper sites. More importantly, unprecedentedly stability of the corresponding Cu catalyst was firstly observed for selective 1,3-butadiene hydrogenation, with both 100% butadiene conversion and 100% butenes selectivity over 120 h of reaction at 75 °C. This study verifies that a simply control the carbonization process of metal organic frameworks can be an effective way to obtain Cu-based catalysts with superior catalytic performance for selective hydrogenation reaction.