Phenolic compounds from mango (M. indica) seed kernels (MSK) var. Sugar were obtained using supercritical CO₂ and EtOH as an extraction solvent. For this purpose, a central composite design was carried out to evaluate the effect of extraction pressure (11–21 MPa), temperature (40–60 °C), and co-solvent contribution (5–15% w/w EtOH) on (i) extraction yield, (ii) oxidative stability (OS) of sunflower edible oil (SEO) with added extract using the Rancimat method, (iii) total phenolics content, (iv) total flavonoids content, and (v) DPPH radical assay. The most influential variable of the supercritical fluid extraction (SFE) process was the concentration of the co-solvent. The best OS of SEO was reached with the extract obtained at 21.0 MPa, 60 °C and 15% EtOH. Under these conditions, the extract increased the OS of SEO by up to 6.1 ± 0.2 h (OS of SEO without antioxidant, Control, was 3.5 h). The composition of the extract influenced the oxidative stability of the sunflower edible oil. By SFE it was possible to obtain extracts from mango seed kernels (MSK) var. Sugar that transfer OS to the SEO. These promissory extracts could be applied to foods and other products.