Staphylococcus aureus is a major cause of subclinical mastitis in dairy cows, and its development of antibiotic resistance has limited treatment efficacy. Essential oils (EOs) are natural products with a wide range of antimicrobial properties that could be used to treat bovine mastitis. This study aims to investigate the antimicrobial activity of EOs against S. aureus isolated from subclinical bovine mastitis cases in the State of São Paulo—Brazil. A total of 14 S. aureus isolates were selected, based on the presence of biofilm-forming genes (icaA, icaD, and bap), and were cultured to a final concentration of 103 CFU.mL−1 for the minimum inhibitory concentration (MIC) and minimum bactericidal concentration (MBC) analysis of five EOs (Citrus aurantium bergamia—bergamot, Copaifera reticulata—copaiba, Foeniculum vulgare—fennel, Zingiber officinale—ginger, and Ocimum basilicum—basil). The chemical compositions of the EOs were characterized using gas chromatography coupled with a mass-selective detector (GC/MSD). Basil and bergamot EOs exhibited the highest antimicrobial activity against S. aureus strains, with mean MIC/MBC values of 1.561 ± 0.223/2.806 ± 0.255 mg.mL−1 and 2.782 ± 0.228/4.396 ± 0.198 mg.mL−1, respectively. The primary compounds in basil EO were methyl-chavicol, linalool, and α-humulene, while bergamot EO predominantly contained linalyl acetate, limonene, and linalool. This research highlights the potential of basil and bergamot EOs as natural antimicrobial agents for treating bovine mastitis caused by S. aureus, offering a potential alternative to traditional antibiotics and contributing to animal welfare and public health. In addition, it emphasizes the need for further studies to validate the long-term effects, optimal dosages, and application methods.