Medicinal plants have been shown to have both chemopreventive and/or therapeutic effects on cancer and other diseases related to oxidative damage. Moringa oleifera Lam., known in the Hausa and Igala languages of Nigeria as "Zogale" and "Gergedi," respectively, and drumstick in English, is a plant that is used both as food and in folkloric medicine in Nigeria and elsewhere. Different parts of the plant were analyzed for polyphenol content as well as in vitro antioxidant potential. The methanol extract of the leaves of M. oleifera contained chlorogenic acid, rutin, quercetin glucoside, and kaempferol rhamnoglucoside, whereas in the root and stem barks, several procyanidin peaks were detected. With the xanthine oxidase model system, all the extracts exhibited strong in vitro antioxidant activity, with 50% inhibitory concentration ($IC_{50}$) values of 16, 30, and 38 ${mu}L$ for the roots, leaves, and stem bark, respectively. Similarly, potent radical scavenging capacity was observed when extracts were evaluated with the 2-deoxyguanosine assay model system, with $IC_{50}$ values of 40, 58, and 72 ${mu}L$ for methanol extracts of the leaves, stem, and root barks, respectively. The high antioxidant/radical scavenging effects observed for different parts of M. oleifera appear to provide justification for their widespread therapeutic use in traditional medicine in different continents. The possibility that this high antioxidant/radical scavenging capacity may impact on the cancer chemopreventive potential of the plant must be considered.