The application of biostimulants to prevent stress-related losses and increase productivity is becoming an increasingly common practice. A study was carried out to determine the effect of the type and methods of biostimulant application and long-term storage on the content of antioxidant compounds determining the processes of the enzymatic browning of carrot roots. The natural seaweed biostimulant Kelpak and synthetic Asahi were applied as part of the study. A quality assessment of carrot roots was carried out directly after harvest and after six months of storage in air at a temperature of 1 °C and a relative humidity of 95%. A single application of Kelpak seaweed extract in the four-leaf phase (2 or 3 dm3 ha−1) proved to be the most effective for reducing oxidative darkening processes (by 33.5%). The oxidative potential was most strongly dependent on the anthocyanin (r = −0.477) and chlorogenic acid (r = −0.474) contents. The concentration of polyphenols (r = 0.836; r = 0.719) and flavonoids (r = 0.671; r = 0.729) had the greatest effect on the antioxidant capacity (FRAP and ABTS respectively). It was observed that pectin, polyphenol, chlorogenic acid, flavonoid and anthocyanin concentrations tended to increase after a single application of Kelpak in a dose of 2 dm3 ha−1. A long-term storage period had a stronger effect on the increase in the oxidation potential than on the antioxidant capacity FRAP value.