Drexel was born in 1826 in Philadelphia to Francis Martin Drexel (1792–1863) and Catherine Hookey (1795–1870). He was the brother of Francis Anthony Drexel, and Joseph William Drexel. He was the uncle of Saint Katharine Drexel.
At the age of 13 he began to work in the banking house founded three years earlier by his father, the Austrian-born American banker Francis Martin Drexel. In 1847 he was named a member of the firm Drexel & Company, the original predecessor of what would become Drexel Burnham Lambert.
After the death of his father in 1863, Drexel closed the bank's Chicago and San Francisco offices and changed the name of its New York branch from Read, Drexel & Co. to Drexel Winthrop. In 1867 he founded a separate Paris-based banking partnership, Drexel, Harjes & Co., with John H. Harjes and Eugene Winthrop.
Three years later, in 1871, at the urging of Junius Spencer Morgan in London, Drexel became the mentor of Junius's troubled son, John Pierpont Morgan of New York, and entered into a new partnership with young Morgan, forming Drexel, Morgan & Co. This new merchant banking partnership, which was based in New York, rather than Philadelphia, served initially as an agent for Europeans investing in the United States. Over the next generation, this partnership assumed the leading role in financing America's railroads and stabilizing and revitalizing Wall Street's chaotic securities markets. The firm created a national capital market for industrial companies— a market that had previously existed only for railroads and canals. To restore investor confidence, Drexel Morgan underwrote the pay of the entire U.S. Army when Congress refused to do so in 1877, bailed out the U.S.Government during the Panic of 1895 and rescued the New York Stock Exchange during the Panic of 1907. With the formation of Drexel, Morgan & Co., Drexel Harjes became the French affiliate of an international banking firm with offices in London, Philadelphia, New York City and Paris that would subsequently become J.P. Morgan & Co..
Two years after Drexel's death in 1893, Drexel, Morgan & Co. was renamed J.P. Morgan & Co., one of the original predecessors of what is today JPMorgan Chase. In 1901, the bank financed the formation of the United States Steel Corporation, the world's first billion-dollar corporation, which took over the business of Andrew Carnegie and other companies.
Drexel married Ellen B. Rozet (1832–1891), the daughter of John Roset (1794–1870) and Mary Ann Laning (1807–1880). Together, they had the following children:Emilie Taylor Drexel (1851–1883), who married Edward Biddle III (born 1851)
Frances Katherine Drexel (1852–1892), who married James William Paul, Jr.
Marie Rozet Drexel (1854–1855), who died young.
Mae E. Drexel (1857–1886), who married Charles T. Stewart
Sarah Rozet "Sallie" Drexel (1860–1929), who married John R. Fell, Sr. (1858–1895), and after his death, married Alexander Van Rensselaer (1850–1933)
Francis Anthony Drexel II (1861–1869)
John Rozet Drexel (1863–1935), who married Alice Gordon Troth (1865–1947)
Anthony Joseph Drexel, Jr. (1864–1934), who married Margarita Armstrong (1867-1948). They divorced in 1917 and in 1918, she married Brinsley FitzGerald (1859-1931)
George William Childs Drexel (1868–1944), who married Mary Stretch Irick (1868–1948).
Drexel died of a heart attack on June 30, 1893 in Karlsbad (in the German-speaking part of Bohemia, Austrian Empire), today Karlovy Vary, Czech Republic, at the age of 66, and was buried in Woodlands Cemetery in Philadelphia.
Through his son, Anthony Joseph Drexel, Jr., his grandchildren included: Margaretta Armstrong Drexel (1885-1952), who married Guy Finch-Hatton, 14th Earl of Winchilsea (1885-1939), Anthony Joseph Drexel II (1887–1946), who married Marjorie Gould (1890–1955), eldest daughter of George Jay Gould I, John Armstrong Drexel (1891-1958), an aviation pioneer, and Louis Clapier Norris Drexel (1896-1962), who married Nancy Grayson.
Through his eldest daughter, he was also the grandfather of Anthony Joseph Drexel Biddle, Sr. (1874–1948) and the great-grandfather of Anthony Joseph Drexel Biddle, Jr. (1897–1961), the United States Ambassador to Czechoslovakia, Poland, and Norway.