From grape cultivation to ripening and harvest timing to processing, each step of the winemaking process can be a critical point when it comes to wine quality and phenolic composition. In this study, the influence of winemaking technology on resveratrol and quercetin content, as well as other polyphenolic compounds, was investigated. Resveratrol is a non-flavonoid polyphenolic stilbene synthesized by grape skin when damaged by infectious diseases or ionizing radiation. Quercetin is a phenol found in grape skins and stems and is produced to protect grapes from UV light damage. Trans-resveratrol and quercetin are known to act as antioxidants, reduce the risk of atherosclerosis and type 2 diabetes, inhibit the growth of cancer cells, and prevent the release of allergic and inflammatory molecules. However, the question was whether red wine could be enriched with these phenols using a co-inoculation winemaking technology. The main new idea was to completely replace the cold maceration process with maceration with the addition of wild yeast (Torulaspora delbrueckii, Td). Maceration with the addition of wild yeast (Td) offers the following advantages over traditional cold maceration: (1) higher concentrations of trans-resveratrol (>35–40%) and quercetin (>35–40%) in the final wine, (2) the new wine has a higher potential for human health, (3) the wine has better aroma and stability due to the higher mannoprotein content, and (4) better energy efficiency in the production process. The study of stability during storage and aging also included derivatives of benzoic acid and hydroxycinnamic acid, piceid, catechin, naringenin, rutin, kaempherol, hesperetin, and anthocyanins. This study found that younger wines had higher phenolic content, while storage of the wine resulted in a decrease in total phenolic content, especially monomeric stilbenes and quercetin. This study represents a small part of the investigation of the influence of non-Saccharomyces yeasts on the phenolic profile of wine, which still requires extensive research with practical application. In addition, non-Saccharomyces yeasts such as Kluyveromyces thermotolerans, Candida stellata, and Metschnikowia pulcherrima could also be used in future studies.