The influence of the technological processes of grape processing on the content of hydroxycinnamic acids and volatile phenolic substances in wine was studied. The method of targeted oxygenation was applied in grape processing technology of the Welschriesling and Chardonnay grape varieties. The content of volatile phenolic substances was determined by gas chromatography, the content of hydroxycinnamic acids by liquid chromatography, and the basic analytical parameters of the wine by FTIR spectrometry. The method of targeted must oxygenation had a statistically significant effect on the content of hydroxycinnamic acids and volatile phenolics in the wine. In all three monitored years (2015–2017), the content of 4-vinylphenol and 4-vinylguajacol in the wine significantly decreased. A significant dependence between the content of hydroxycinnamic acids and volatile phenolics was found. The experiment showed that a 1% increase in the content of hydroxycinnamic acids in the Chardonnay variety could result in an average increase in the content of monitored volatile phenolics by 3.6% (3 years’ data). Naturally reducing the content of hydroxycinnamic acids, with the application of technological processes, eliminated the oxidative processes during wine maturation. Sensory undesirable volatile phenolic substances were consequently formed in lower quantities, and there was no negative impact on the favourable sensory properties of wine. It was not necessary to use the polyvinylpolypyrrolidone adsorbents.