Andean potatoes (Solanum tuberosum L. ssp. andigena) are a good source of dietary antioxidant polyphenols. We have previously demonstrated that polyphenol extracts from Andean potato tubers exerted a dose-dependent cytotoxic effect in human neuroblastoma SH-SY5Y cells, being skin extracts more potent than flesh ones. In order to gain insight into the bioactivities of potato phenolics, we investigated the composition and the in vitro cytotoxic activity of total extracts and fractions of skin and flesh tubers of three Andean potato cultivars (Santa María, Waicha, and Moradita). Potato total extracts were subjected to liquid–liquid fractionation using ethyl acetate solvent in organic and aqueous fractions. We analyzed both fractions by HPLC-DAD, HPLC-ESI-MS/MS, and HPLC-HRMS. Results corroborated the expected composition of each fraction. Organic fractions were rich in hydroxycinnamic acids (principally chlorogenic acid isomers), whereas aqueous fractions contained mainly polyamines conjugated with phenolic acids, glycoalkaloids, and flavonoids. Aqueous fractions were cytotoxic against SH-SY5Y cells and even more potent than their respective total extracts. Treatment with a combination of both fractions showed a similar cytotoxic response to the corresponding extract. According to correlation studies, it is tempting to speculate that polyamines and glycoalkaloids are crucial in inducing cell death. Our findings indicate that the activity of Andean potato extracts is a combination of various compounds and contribute to the revalorization of potato as a functional food.