Weather conditions affect vine development and grape composition, although the response may be different depending on the variety and altitude. Under scenarios of climate change, the knowledge of the relationships between grape composition and climate is therefore important to know the suitability of a given cultivar. In this research, the variability of the grape composition of two red varieties with different phenological timings (Vitis vinifera L. cv. Tempranillo and Graciano) cultivated in Rioja DOCa (Spain) and the climatic variables that have a higher effect on the response of each of them were analysed. Grape composition of both cultivars at technical maturity (considered when a probable volumetric alcoholic degree (PVAD) = 13 º was reached) was analysed during the period 2008–2020 in areas located at different elevations and related to the weather conditions recorded in those areas. The results show the effect of temperature and water availability in different periods during the growing cycle on grape composition and the potential benefits of cultivating at a higher elevation, under higher water availability. The anthocyanins were affected by the maximum temperatures recorded in the period before veraison and during ripening (period veraison to technical maturity), decreasing their concentrations with increasing temperatures. In addition, higher water availability gave rise to an increase in acidity. Graciano is shown as a variety more suitable than Tempranillo to be cultivated under warmer conditions.