Kennelly was born in Colaba, in South Mumbai, India and was educated at University College School in London. He was the son of an Irish naval officer Captain David Joseph Kennelly (1831-1907) and Catherine Gibson Heycock (1839-1863). His mother died when he was three years old. Afterwards, in 1863, his father retired from the navy and later Arthur and his father returned to England. In 1878, his father remarried to Ellen L.Vivian and moved the family to Sydney, Nova Scotia on the island of Cape Breton when he took over the Sydney and Louisbourg Coal and Railway Company Limited. By his father's second marriage, Arthur gained four half siblings, Zaidia Kennelly in 1881, David J.Kennelly, Jr. in 1882, Nell K.Kennelly in 1883 and Spencer M.Kennelly in 1885.
Joined Thomas Edison's West Orange laboratory in Dec 1887, staying until March 1894. He formed a consulting firm in electrical engineering with Edwin Houston. Together, they wrote Alternating Electric Currents (1895), Electrical Engineering leaflets (1896), Electric arc lighting (1902).
In 1893, during his research in electrical engineering, he presented a paper on "Impedance" to the American Institute of Electrical Engineers (AIEE). He researched the use of complex numbers as applied to Ohm's Law in alternating current circuit theory. In 1902, he investigated the ionosphere's radio spectrum's electrical properties. He was a professor of electrical engineering at Harvard University from 1902-1930 and jointly at the Massachusetts Institute of Technology from 1913-1924. One of his PhD students was Vannevar Bush.
Kennelly was the recipient of the awards of many nations, including the IEE Institution Premium (1887), the Franklin Institute Howard Potts Gold Medal (1917), the Cross of a Chevalier of the Légion d'honneur of France and the AIEE, now IEEE, Edison Medal (1933) "For meritorious achievements in electrical science, electrical engineering and the electrical arts as exemplified by his contributions to the theory of electrical transmission and to the development of international electrical standards." He was awarded the IRE Medal of Honor in 1932, "For his studies of radio propagation phenomena and his contributions to the theory and measurement methods in the alternating current circuit field which now have extensive radio application." He was an active participant in professional organizations such as the Society for the Promotion of the Metric System of Weights and Measures, the Illuminating Engineering Society and the U.S. National Committee of the International Electrotechnical Commission, and also served as the president of both the AIEE and the Institute of Radio Engineers, IRE, during 1898-1900 and 1916, respectively. Kennelly died in Boston, Massachusetts on 18 June 1939.