Elisha Baxter was born in Rutherford County, North Carolina. Baxter received an appointment to the United States Military Academy but his father would not allow him to attend.
In 1852 Baxter moved to Arkansas and opened a business that soon failed. He was elected to the Arkansas Senate in 1854. He studied law and, in 1856, was admitted to the Arkansas bar. He was reelected to the Senate in 1858 and served until 1860.
At the start of the American Civil War Baxter refused to fight for the Confederacy and attempted to flee to Missouri. He was captured and tried for treason. He escaped north and joined the 4th Arkansas Mounted Infantry (USA) and served as Colonel in that regiment.
In 1864 he was appointed as Chief Justice of the Arkansas Supreme Court in the reconstruction government. From 1868 to 1872 he served as a judge on the 3rd Circuit Court.
In 1872 he was elected Governor of Arkansas over Joseph Brooks in a controversial election that led to the Brooks-Baxter War. Baxter was physically removed from the governor's office by Brooks and state militia loyal to him. Baxter was not restored to the governorship until a month later.
During his term a new constitution was formed which shortened his term and returned voting rights to ex-Confederates. Baxter declined to accept the 1874 nomination for governor and was the last Republican governor to be elected in Arkansas until Winthrop Rockefeller in 1967.
After leaving office Baxter returned to his farm near Batesville, Arkansas. He ran for a position in the Arkansas House of Representatives in 1878 but was unsuccessful.
Elisha Baxter is buried at Oaklawn Cemetery in Batesville, Arkansas.
Baxter County, Arkansas was named after Elisha Baxter.