Grapevines worldwide are grafted onto Vitis spp. rootstocks in order to improve their tolerance to biotic and abiotic stresses. Thus, the response of vines to drought is the result of the interaction between the scion variety and the rootstock genotype. In this work, the responses of genotypes to drought were evaluated on 1103P and 101-14MGt plants, own-rooted and grafted with Cabernet Sauvignon, in three different water deficit conditions (80, 50, and 20% soil water content, SWC). Gas exchange parameters, stem water potential, root and leaf ABA content, and root and leaf transcriptomic response were investigated. Under well-watered conditions, gas exchange and stem water potential were mainly affected by the grafting condition, whereas under sever water deficit they were affected by the rootstock genotype. Under severe stress conditions (20% SWC), 1103P showed an “avoidance” behavior. It reduced stomatal conductance, inhibited photosynthesis, increased ABA content in the roots, and closed the stomata. The 101-14MGt maintained a high photosynthetic rate, limiting the reduction of soil water potential. This behavior results in a “tolerance” strategy. An analysis of the transcriptome showed that most of the differentially expressed genes were detected at 20% SWC, and more significantly in roots than in leaves. A core set of genes has been highlighted on the roots as being related to the root response to drought that are not affected by genotype nor grafting. Genes specifically regulated by grafting and genes specifically regulated by genotype under drought conditions have been identified as well. The 1103P, more than the 101-14MGt, regulated a high number of genes in both own-rooted and grafted conditions. This different regulation revealed that 1103P rootstock readily perceived the water scarcity and rapidly faced the stress, in agreement with its avoidance strategy.