Potato (Solanum tuberosum L.) is an important food and vegetable crop worldwide. In recent years, the arid environment resulting from climate change has caused a sharp decline in potato yield. To clarify the effect of drought priming at the seedling stage on the tolerance of potato plants to drought stress during tuber expansion, we conducted a pot experiment to investigate the physiological response of the plants generated from seed potatoes of the variety ‘Favorita’ to varied water supply conditions: normal water supply at the seedling stage (control), normal water supply at the seedling stage and drought stress at the mid-tuber-expansion stage (non-primed), and drought priming at the seedling stage plus drought stress at the mid-tuber-expansion stage (primed). Drought priming resulted in an increase in the number of small vascular bundles in potato plants compared to non-primed plants. It also altered the shape and density of stomata, enhancing water use efficiency and reducing whole-plant transpiration. The primed plants maintained the basal stem cambium for a longer time under drought stress, which gained an extended differentiation ability to generate a greater number of small vascular bundles compared to non-primed plants. Drought priming increased the amount and rate of dry matter translocation, and so reduced the adverse effects on tubers of potato under drought stress. Therefore, drought priming at the seedling stage improved the photosynthetic performance and yield, and probably enhanced the drought tolerance of potato.