As key regulators of the Jasmonates (JAs) signal transduction pathway, JAZ protein, and MYC transcription factors are imperative for plant response to external environmental changes, growth, and development. In this study, 18 StJAZs and 12 StMYCs were identified in potatoes. Their chromosomal position, phylogenetic development, gene structure, and promoter cis-acting parts of the StJAZ genes were analyzed. In addition, Protein–Protein Interaction (PPI) network analysis of StJAZ and StMYC gene families and yeast two-hybrid assay demonstrated that five StMYCs can interact with 16 StJAZs, which provides new insights into the operation mechanism of StJAZs and StMYCs in JA signal response. Moreover, we explored the expression profiles of StJAZs and StMYCs genes in different tissues and during abiotic stresses by RNA-seq data. Based on the PPI network and transcriptome data, the genes StJAZ11, StJAZ16, and StMYC6 were chosen for further qRT-PCR study under salt or mannitol treatment. Under mannitol-induced drought or salinity treatment, the expression patterns of StMYC6, StJAZ11, and StJAZ16 were different, indicating that the JAZ protein and MYC transcription factor may be engaged in the response of potatoes to abiotic stress, which opened up a new research direction for the genetic improvement of potatoes in response to environmental stress.