Born near Warrenton, North Carolina, Bragg attended the local academy at Warrenton, and was graduated from the University of North Carolina at Chapel Hill in 1824. He studied law. He was admitted to the bar in 1830 and commenced practice in Warrenton. He served as member of the State house of commons of North Carolina 1830-1834. He moved to Mobile, Alabama, in 1836 and continued the practice of law. He was appointed judge of the tenth judicial circuit in 1842. He served as member of the State house of representatives.
Bragg was elected as a Democrat to the Thirty-second Congress (March 4, 1851-March 3, 1853). He declined to be a candidate for reelection in 1852. He resumed the practice of his profession. He built a home in Mobile in 1855 that is known today as the Bragg-Mitchell Mansion. He served as delegate from Mobile to the State constitutional convention in 1861. He died in Mobile, Alabama, August 10, 1878. He was interred in Magnolia Cemetery.