Prunus persica L. Batsch, or peach, is one of the most important crops and it is widely established in irrigated arid and semi-arid regions. However, due to variations in the climate and the increased aridity, drought has become a major constraint, causing crop losses worldwide. The use of drought-tolerant rootstocks in modern fruit production appears to be a useful method of alleviating water deficit problems. However, the transcriptomic variation and the major molecular mechanisms that underlie the adaptation of drought-tolerant rootstocks to water shortage remain unclear. Hence, in this study, high-throughput sequencing (RNA-seq) was performed to assess the transcriptomic changes and the key genes involved in the response to drought in root tissues (GF677 rootstock) and leaf tissues (graft, var. Catherina) subjected to 16 days of drought stress. In total, 12 RNA libraries were constructed and sequenced. This generated a total of 315 M raw reads from both tissues, which allowed the assembly of 22,079 and 17,854 genes associated with the root and leaf tissues, respectively. Subsets of 500 differentially expressed genes (DEGs) in roots and 236 in leaves were identified and functionally annotated with 56 gene ontology (GO) terms and 99 metabolic pathways, which were mostly associated with aminobenzoate degradation and phenylpropanoid biosynthesis. The GO analysis highlighted the biological functions that were exclusive to the root tissue, such as “locomotion,” “hormone metabolic process,” and “detection of stimulus,” indicating the stress-buffering role of the GF677 rootstock. Furthermore, the complex regulatory network involved in the drought response was revealed, involving proteins that are associated with signaling transduction, transcription and hormone regulation, redox homeostasis, and frontline barriers. We identified two poorly characterized genes in P. persica: growth-regulating factor 5 (GRF5), which may be involved in cellular expansion, and AtHB12, which may be involved in root elongation. The reliability of the RNA-seq experiment was validated by analyzing the expression patterns of 34 DEGs potentially involved in drought tolerance using quantitative reverse transcription polymerase chain reaction. The transcriptomic resources generated in this study provide a broad characterization of the acclimation of P. persica to drought, shedding light on the major molecular responses to the most important environmental stressor.