The yield of sweet potato [Ipomoea batatas (L.) Lam] can be easily threatened by drought stress. Typically, early stages like the seedling stage and tuber-root expansion stage are more vulnerable to drought stress. In this study, a highly drought-tolerant sweet potato cultivar “WanSu 63” was subjected to drought stress at both the seedling stage (15 days after transplanting, 15 DAT) and the tuber-root expansion stage (45 DAT). Twenty-four cDNA libraries were constructed from leaf segments and root tissues at 15 and 45 DAT for Next-Generation Sequencing. A total of 663, 063, and 218 clean reads were obtained and then aligned to the reference genome with a total mapped ratio greater than 82.73%. A sum of 7119, 8811, 5463, and 930 differentially expressed genes were identified from leaves in 15 days (L15), roots in 15 days (R15), leaves in 45 days (L45), and roots in 45 days (R45), respectively, in drought stress versus control. It was found that genes encoding heat shock proteins, sporamin, LEA protein dehydrin, ABA signaling pathway protein gene NCED1, as well as a group of receptor-like protein kinases genes were enriched in differentially expressed genes. ABA content was significantly higher in drought-treated tissues than in the control. The sweet potato biomass declined sharply to nearly one-quarter after drought stress. In conclusion, this study is the first to identify the differentially expressed drought-responsive genes and signaling pathways in the leaves and roots of sweet potato at the seedling and root expansion stages. The results provide potential resources for drought resistance breeding of sweet potato.