Anthraquinones are of significant interest due to their biological activity, coloring properties, and synthetic applications. Here, we describe a mild and convenient method for modification of 1-ethynyl-4-hydroxyanthraquinone that was obtained from the Sonogashira cross-coupling reaction of 1-hydroxy-4-iodoanthraquinone with alkynes. The copper(I) catalyzed one-pot three-component reaction (A3-coupling) of the new 1-ethynyl-4-hydroxyanthraquinone with secondary amines and formaldehyde was the main approach for the synthesis of nitrogen-substituted 1-[3-(amino)prop-1-ynyl]-4-hydroxyanthraquinones. The influence of different substituents in the amine on reaction rate and yield has been evaluated. The cytotoxicity of 1-ethynyl-4-hydroxyanthraquinones was assessed using the conventional MTT assay. Among all compounds synthesized, anthraquinone-propargylamine derivatives 28, 29, 30, and 34 possess the most promising cytotoxic potential towards glioblastoma cancer cells; compounds 14 and 19 shown selectivity towards the prostate cancer cells DU-145, and 18, 24 — on breast cancer cell line MCF-7. The grown inhibition on these cancer cells by compounds 18 and 24 was comparable to those of standard drug Doxorubicin. Molecular modeling of new anthraquinone derivatives in DNA G-quadruplex binding sites was performed to help understand the observed SAR trends.