Organic acids represent naturally occurring compounds that are found in many types of food and beverages, with important functions in defining products’ final quality. Their proportions in wine are dependent on grape composition and winemaking conditions (temperature, pH levels, oxygen, and carbon dioxide concentration). Therefore, this article studied the influence of different fermentation conditions (200 hL tanks vs. 50 L glass demijohns) and various yeasts on the evolution of the main organic acids during alcoholic fermentation of “Aligoté” wines. The fermentation lasted 22 days and samples were collected daily. Laboratory analyses were quantified according to the International Organization of Vine and Wine recommendations. High-performance liquid chromatography for the identification and quantification of organic acids was used. The data showed the important effect of winemaking conditions on sugar consumption, density or acidity values, and sensory characteristics. Significant differences in organic acid concentrations (especially for tartaric acid) were obtained for all variants, depending on the fermentation conditions, inoculated yeast and the sampling moment. The quantities of most of the identified organic acids were generally significantly increased when glass vessels were used, compared to those fermented in tanks. Most organic acids concentrations were favored by lower pH and showed higher values at lower temperatures.