Abiotic stresses comprise all nonliving factors, such as soil salinity, drought, extreme temperatures, and metal toxicity, posing a serious threat to agriculture and affecting the plant production around the world. Peanut (Arachis hypogaea L.) is one of the most important crops for vegetable oil, proteins, minerals, and vitamins in the world. Therefore, it is of importance to understand the molecular mechanism of peanut against salt stress. Six transcriptome sequencing libraries including 24-hour salt treatments and control samples were constructed from the young leaves of peanut. A comprehensive analysis between two groups detected 3,425 differentially expressed genes (DEGs) including 2,013 upregulated genes and 1,412 downregulated genes. Of these DEGs, 141 transcription factors (TFs) mainly consisting of MYB, AP2/ERF, WRKY, bHLH, and HSF were identified in response to salinity stress. Further, GO categories of the DEGs highly related to regulation of cell growth, cell periphery, sustained external encapsulating structure, cell wall organization or biogenesis, antioxidant activity, and peroxidase activity were significantly enriched for upregulated DEGs. The function of downregulated DEGs was mainly enriched in regulation of metabolic processes, oxidoreductase activity, and catalytic activity. Fourteen DEGs with response to salt tolerance were validated by real-time PCR. Taken together, the identification of DEGs' response to salt tolerance of cultivated peanut will provide a solid foundation for improving salt-tolerant peanut genetic manipulation in the future.