Stevia rebaudiana Bertoni is a perennial shrub belonging to the Asteraceae family. The leaves contain a mixture of steviol glycosides with extraordinary sweetening properties, among which the most important are stevioside and rebaudioside A. These components have a high sweetening power, which is about 300 times that of sucrose, and a negligible calorie content. However, their extraction and purification are not easy. In this paper, the extraction technique under cyclic pressure, known as rapid solid-liquid dynamic extraction (RSLDE), was compared using a Naviglio extractor (NE) with conventional maceration. The aim was to identify an efficient and economically viable method for obtaining high amounts of steviol glycosides in a short time. Furthermore, a numerical model was set up for the solid-liquid extraction process of value-added compounds from natural sources. Several parameters must be evaluated in relation to the characteristics of the parts of the plant subjected to extraction. Therefore, since diffusion and osmosis are highly dependent on temperature, it is necessary to control the temperature of the extraction system. On the other hand, the final aim of this work was to provide a scientific and quantitative basis for RSLDE. Therefore, the results obtained from stevia extracts using the corresponding mathematical model allowed hypothesizing the application of this model to the extraction processes of other vegetable matrices.