Jonsson began his career in 1922 at Alcoa as a rolling mill apprentice, and advanced to the position of manufacturing superintendent of an Alcoa subsidiary, the Aluminum Index Company. He held the position from 1923 to 1927. In 1927 he left Alcoa and entered the auto business attempting a Pontiac dealership, returning to Alcoa in 1929 as a sales engineer. During Jonsson's last year at Alcoa, J. Clarence Karcher the husband of Mrs. Jonsson's cousin, occasionally asked Jonsson's assistance with expediting materials orders for his start-up company, Geophysical Service Incorporated a pioneering provider of seismic exploration services to the petroleum industry. In June 1930 Karcher offered Jonsson a job managing the manufacture of seismic instruments at Karcher's company lab in Newark, New Jersey, and in July Jonsson again left Alcoa. In 1934, Jonsson and his family moved to Dallas, Texas. Jonsson held the position of secretary of Geophysical Service, which gave him responsibility for all of the accounting, banking, international, legal, manufacturing, personnel, purchasing, and warehousing activities of the company. Jonsson held this position until 1939. In 1939, the company reorganized as Coronado Corp., an oil company with Geophysical Service Inc, now as a subsidiary. In 1939, Jonsson became secretary-treasurer. On 6 December 1941, Jonsson along with three other GSI employees, Eugene McDermott, Cecil H. Green, and H.B. Peacock purchased GSI. In 1942, he became vice president and treasurer. He held this position until 1951. During World War II, GSI built electronics for the United States Army Signal Corps and the Navy. After the war GSI continued to produce electronics. The rugged nature of equipment for the oil industry and of military equipment were similar and thus continued expansion into military contracts was a natural progression. In 1951, the company changed its name to Texas Instruments; GSI becoming a wholly owned subsidiary of the new company. Jonsson became president of Texas Instruments, a position he held until 1958. Jonsson was elected chairman of the board in 1958 and held this position until 1966. He became honorary chairman in 1966 through 1977.
Civic activities
Jonsson was a strong advocate for education, serving or leading on the boards of many educational institutions and created alliances to improve local educational facilities. Jonsson was a founder of the Southwest Center for Advanced Studies, which became the University of Texas at Dallas in 1969. In 1964, shortly after the assassination in Dallas of President John F. Kennedy, Jonsson became mayor of Dallas, Texas. He worked to improve morale and the image of the city. He was reelected three times. Jonsson pushed through a $175 million bond that financed a new city hall, the Dallas Convention Center and the Dallas Central Library. He was a central leader in the development of Dallas-Fort Worth International Airport.