The consumption of sugar-sweetened beverages has been related with the risk of cardiovascular diseases and other pathophysiological situations, such as obesity or diabetes mellitus. Given the increasing awareness on this fact, food industries are developing new products to reduce the amount of added sugar in development of food products development. Accordingly, in the present work, new functional beverages, constituting a dietary source of bioactive phenolics and supplemented with stevia or sucrose, were designed in order to study the influence of the sweetener during processing and shelf-life. This study is of critical for the informed selection of the sweetener based on its effect on the final phytochemical profile of beverages, especially taking into consideration that there are no previous studies on Stevia rebaudiana. Physicochemical features and phytochemical composition, as well as stability of the different beverages concerning these parameters, were evaluated for 90 days during storage under different conditions (refrigeration (4 °C) and room temperature (25 °C) under light or darkness conditions). Physicochemical parameters (pH, titratable acidity, total soluble solids, and color) did not display statistically significant differences between beverages. Storage temperature was the greatest determinant affecting the stability of all the analyzed bioactive compounds (vitamin C, anthocyanins, and flavanones). The main difference between sweeteners was observed in flavanones, which exhibited a higher loss during storage under day light conditions when stevia was added instead of sucrose. In addition, the juices’ colors were rather stable, keeping a reddish coloration and natural appearance throughout the shelf life. Hence, stevia could be considered as an alternative sweetener by the beverage industry.