Chlorophyll (CHL) is present in many plant organs, and its metabolism is strongly regulated throughout plant development. Understanding the fate of CHL in senescent leaves or during fruit ripening is a complex process. The stay-green (SGR) protein has been shown to affect CHL degradation. In this study, we used the conserved sequences of STAY-GREEN domain protein (NP_567673) in Arabidopsis thaliana as a probe to search SGR family genes in the genome-wide melon protein database. Four candidate SGR family genes were identified in melon (Cucumis melo L. Hetao). The phylogenetic evolution, gene structure, and conserved motifs were subsequently analyzed. In order to verify the function of CmSGR genes in CHL degradation, CmSGR1 and CmSGR2 were transiently overexpressed and silenced using different plasmids in melon. Overexpression of CmSGR1 or CmSGR2 induced leaf yellowing or fruit ripening, while silencing of CmSGR1 or CmSGR2 via RNA interference delayed CHL breakdown during fruit ripening or leaf senescence compared with the wild type. Next, the expression profile was analyzed, and we found that CmSGR genes were expressed ubiquitously. Moreover, CmSGR1 and CmSGR2 were upregulated, and promoted fruit ripening. CmSGR3 and CmSGR4 were more highly expressed in leaves, cotyledon, and stem compared with CmSGR1 or CmSGR2. Thus, we conclude that CmSGR genes are crucial for fruit ripening and leaf senescence. CmSGR protein structure and function were further clarified to provide a theoretical foundation and valuable information for improved performance of melon.