Born in Tiflis (modern Tbilisi), Mantashev spent most of his childhood in Tabriz where his father was involved in the cotton and textile trade. Being the only son, he was involved in his father's business affairs early on. In 1869, he moved to Manchester (known as Cottonopolis in the 19th century), a major center of cotton and textile processing industries from where he helped ship goods to his father in Tabriz. Mantashev's stay in Manchester played an important role in the development of his character. Not only did he learn the secrets and crafts of the textile industry in Manchester, but he also delved into the intricacies of European business and English culture. During this period he learned the English, French, and German languages. In 1872, Mantashev returned to Tiflis with his father. On the first floor of the Hotel Caucasus, located in Erivansky Square, the Mantashevs opened a cotton store, then another one, eventually becoming fully engaged in the wholesale textile trade. After his father's death in 1887, Alexander purchased most of the shares of the Tiflis Central Commercial Bank – thus becoming its principal shareholder, and then the Chairman of the Board of the Bank. The bank was involved in almost every aspect of trade in the Caucasus. Incidentally, Tifkombank was the only financial institution in the Caucasus whose shares traded on the Saint Petersburg Stock Exchange. In the early 1890s, Alexander was already a 1st guild merchant and a Speaker of the Tiflis duma. It was then that he became interested in a new business venture when he was looking into the prospects of Baku oil.

Along with twelve like-minded people he founded the "Armenian Charitable Society in the Caucasus". He donated 300,000 rubles towards the building of the Nersessian spiritual academy. He donated 250,000 rubles to the holy town of Echmiadzin for the building of the residence of the Catholicos of Armenia and of All Armenians (construction completed in 1914). Mantashev hand-picked fifty talented young Armenians and sent over two hundred to study at the best universities in Russia and Europe. Among them were the famous Armenian composer Komitas and the controversial Communist revolutionary Stepan Shahumyan as well as others who became famous later on. The most famous donation made by Mantashev remains the Armenian Church of St. John the Baptist in Paris on Jean Goujon Street. He explained that he chose Paris because that's the city where he sinned most. During its construction in 1904, Mantashev spent 1,540,000 francs. For this act, the President of France gave Alexander Mantashev the Order of the Legion of Honor.