Braxton was born on September 5, 1836, in Norfolk, the son of Carter Moore Braxton and his third wife, Elizabeth Teagle Mayo Braxton. Elliott Muse Braxton, who served in the U.S. House of Representatives after the Civil War, was his elder half brother. Carter Moore Braxton lived in Norfolk and in King and Queen County until his father died in 1847, after which he lived in Fredericksburg. He was educated at Hanover Academy and prepared for a career in civil engineering. By the time the Civil War broke out in 1861, Braxton was chief engineer in charge of construction for the Fredericksburg and Gordonsville Railroad.

Braxton had a distinguished career in the Confederate army. He became a captain in the Fredericksburg Artillery on May 8, 1861. His company, known as Braxton's Battery, saw action in the Seven Days' Battles and at Cedar Mountain, the Second Battle of Manassas, Chantilly, Antietam, and Fredericksburg (all in 1862). When the Army of Northern Virginia reorganized its artillery, Braxton was promoted to major on April 4, 1863, with the commission to date from March 2. After fighting at Chancellorsville and distinguishing himself at Gettysburg and Mine Run (all in 1863), he was promoted on March 14, 1864, to lieutenant colonel to rank from February 27, 1864. He served at the battles of the Wilderness (1864) and of Cold Harbor (1864) and saw action in the Shenandoah Valley in 1864 and at Chaffin's Farm near Richmond early in 1865. Braxton fought at Hatcher's Run and Five Forks in the last days of the war and surrendered with the Army of Northern Virginia at Appomattox Court House April 9, 1865. Although often in the thick of the fighting and described by one later account as having had seven horses killed under him, he was never wounded. After the war Braxton prepared and published a Map of the Battle Field of Fredericksburg, Explained by Extracts from Official Reports (1866).

Braxton married Fanny Page Hume in Orange County on February 16, 1865, but she died in Richmond on June 16 of that year. He returned to Fredericksburg and to his career as a civil engineer, first in overseeing the repair of the Aquia Creek bridge of the Richmond, Fredericksburg, and Potomac Railroad, and briefly in 1869 as Fredericksburg city surveyor. In June 1866 Braxton became president of the Fredericksburg and Gordonsville Railroad, which he extended during the following decade to Orange as the Potomac, Fredericksburg, and Piedmont Railroad. He also supervised the surveying and construction of the Fredericksburg and Alexandria Railroad, which later became part of the Richmond, Fredericksburg, and Potomac Railroad. Braxton married Nannie Clementina Alsop, of Fredericksburg, on March 26, 1868, and they had nine daughters.

In 1881 Braxton directed the completion of the Chesapeake and Ohio Railway line for the few miles from Lee Hall into Newport News. He moved to the latter town, where from 1881 to 1889 he was in charge of maintenance for the C&O. In July 1889 Braxton began supervision of the massive excavations for the Chesapeake Dry Dock and Construction Company, which changed its name within the year to the Newport News Shipbuilding and Dry Dock Company. He formed his own engineering construction firm, Braxton, Chandler, and Marye, and also developed real estate as president of the Central Land Company and of the Newport News Land and Development Company. Braxton was founding president of the Newport News Street Railway Company, which began with horse- drawn carriages but soon converted to electricity. In 1891 he became vice president of the First National Bank of Newport News, and he was also vice president of the Newport News Gas Company.

Braxton was active in the First Baptist Church of Newport News and from 1887 until his death was commander of the McGruder Camp of the United Confederate Veterans. Carter Moore Braxton died of Bright's disease at his home in Newport News on May 27, 1898, and was buried in the local Greenlawn Cemetery.