Bjarni studied constitutional law and became professor at the University of Iceland at the age of only 24. He was elected to the city council in Reykjavík in 1934 as a member of the Independence Party and from 1940 to 1947 was mayor of the city.
In 1947 he became Foreign Minister and served in various posts in cabinets until 1956. His tenure included the establishment of Iceland as a founding member of NATO in 1949. In 1956, when the left-wing parties formed a coalition government, Bjarni, out of office, became editor of Morgunblaðið, a leading conservative newspaper.
In 1959, when the Independence Party formed a coalition government with the Social Democrats, Bjarni became Minister of Justice. Two years later he was elected chairman of the Independence Party and in 1963 he took over from Ólafur Thors as Prime Minister. He served in this position until his tragic death, which was caused by a fire at a government summer house at Thingvellir; his wife and grandson also perished in the blaze.