Trichloroanisole (TCA) in wine results in a sensory defect called “cork taint”, a significant problem for the wine industry. Wines can become contaminated by TCA absorption from the atmosphere through contaminated wood barrels, cork stoppers, and wood pallets. Air-depleted solvent-impregnated (ADSI) cork powder (CP) was used to mitigate TCA in wines. The ADSI CP (0.25 g/L) removed 91% of TCA (6 ng/L levels), resulting in an olfactory activity value of 0.14. A Freundlich isotherm described ADSI CP TCA adsorption with irreversible adsorption and a KF = 33.37. ADSI CP application had no significant impact on the phenolic profile and chromatic characteristics of red wine. Using headspace sampling with re-equilibration, an average reduction in the volatile abundance of 29 ± 15%, 31 ± 19%, and 37 ± 24% was observed for the 0.10, 0.25, and 0.50 g/L ADSI CP, respectively. The alkyl esters and acids were the most affected. The impact observed was much lower when using headspace sampling without re-equilibration. Isoamyl acetate, ethyl hexanoate, ethyl hexanoate, and ethyl decanoate abundances were not significantly different from the control wine and 0.25 g/L ADSI CP application. Thus, ADSI CP can be a new sustainable fining agent to remove this “off-flavor” from wine, with a reduced impact on the wine characteristics.