The present study tested the effect of a slight increase in pressure (from 0 to 1 bar) during the fermentation on the wine aroma profile. Fermentations were carried out with a commercial dry yeast on Sangiovese juice in the absence of berry skins. The wine samples fermented under slight overpressure conditions were found to be significantly different from the control samples produced at atmospheric pressure in relation to several chemical compounds. Concentrations of many esters (i.e., isoamyl acetate, ethyl acetate, ethyl hexanoate, hexyl acetate, ethyl dodecanoate, and ethyl tetradecanoate), and acids (i.e., hexanoic acid, octanoic acid, and decanoic acid) increased, while concentrations of two acids (i.e., isobutyric and isovaleric acid) decreased. These differences, notably the higher concentration of esters, are usually associated with a more intense fruity attribute. Triangular sensory tests revealed that the significant chemical differences were also perceivable; hence, introducing a slight pressure increase during the alcoholic fermentation could be a useful tool in managing the aroma profile of wine.