This study aimed to characterize the physiological and structural responses of potted one-year-old olive trees belonging to two olive cultivars—‘Nocellara del Belice’ and ‘Cerasuola’—exposed to prolonged drought under greenhouse conditions. Two irrigation treatments based on evapotranspiration (ET) were imposed for 69 days, i.e., well-watered (WW, 100% ET) and drought-stressed (DS, 10−30% ET). Leaf stomatal conductance (gs), stem water potential (Ψstem), transpiration (E), photosynthetic capacity (Amax), water use efficiency (WUE), stem (Kstem) and root (Kroot) hydraulic conductance, trunk diameter variations (TDV), and leaf patch attenuated pressure fluctuations (pp, a proxy of the inverse of leaf turgor pressure) were measured in WW and DS trees at different stages of the experiment. Leaf gs did not significantly differ between cultivars under DS, whereas differences in Ψstem only became significant at the end of prolonged drought, when ‘Nocellara del Belice’ experienced Ψstem < −4 MPa. ‘Cerasuola’ trees expressed the best WUE under drought, although they were more susceptible to photoinhibition under optimal plant water status. Both cultivars tended to increase their Kstem at the end of the drought period. A marked reduction in Kroot occurred in ‘Cerasuola’ plants after prolonged drought; however, a similar mechanism was not observed in ‘Nocellara del Belice’. The ratio between Kstem and Kroot exponentially increased towards the end of the prolonged drought period in both cultivars, but more markedly in ‘Cerasuola’. TDV and pp trends suggested that ‘Cerasuola’ plants keep better plant water status under severe drought compared to ‘Nocellara del Belice’ by maintaining high leaf turgor and reduced trunk diameter fluctuations. These responses may be related to reduced cell wall elasticity and xylem vessel size and/or wall thickness—drought avoidance mechanisms. The Kstem/Kroot ratio can serve as an indicator of drought stress avoidance mechanisms to compare genotype-specific responses to drought stress.