The neem tree, Azadirachta indica, belongs to the Meliaceae family, and its use in the treatment of medical disorders from ancient times to the present in the traditional medical practices of Asia, Africa and the Middle East is well-documented. Neem oil, extracted from the seeds of the fruit, is widely used, with promising medicinal benefits. Azadiradione, a principal antioxidant component of the seeds of A. indica, is known to reduce oxidative stress and has anti-inflammatory effects. To directly measure the antioxidant ability of neem oil, we used Rotating Ring Disk Electrode (RRDE) hydrodynamic voltammetry to quantify how it can scavenge superoxide radical anions. The results of these experiments show that neem oil is approximately 26 times stronger than other natural products, such as olive oil, propolis and black seed oil, which were previously measured using this method. Next, computational Density Functional Theory (DFT) methods were used to arrive at a mechanism for the scavenging of superoxide radical anions with azadiradione. Our work indicates that azadiradione is an effective antioxidant and, according to our DFT study, its scavenging of the superoxide radical anion occurs through a reaction mechanism in which azadiradione mimics the antioxidant action of superoxide dismutase (SOD). In this mechanism, analogous to the SOD enzymatic reaction, azadiradione is regenerated, along with the production of two products: hydrogen peroxide and molecular oxygen. This antioxidant process provides an explanation for azadiradione’s more general and protective biochemical effects.