About 1763 Elliot and his brother Hugh were sent to Paris, where their studies were supervised by the renowned Scottish philosopher David Hume, and where they became intimate with Honoré Mirabeau. Having passed the winters of 1766 and 1767 at the University of Edinburgh, Gilbert entered Christ Church, Oxford, and on quitting the university he was called to the bar. In 1776 he entered parliament as an independent Whig MP for Morpeth. He became very friendly with Edmund Burke, whom he helped in the attack on Warren Hastings and Sir Elijah Impey, and on two occasions was an unsuccessful candidate for the office of Speaker.
In 1794 Elliot was appointed to govern Corsica, and in 1797 he assumed the additional names of Murray-Kynynmound and was created Baron Minto. From 1799 to 1801 he was envoy-extraordinary to Vienna, and having been for a few months president of the board of control he was appointed governor-general of India at the end of 1806. He governed with great success until 1813, during which he expanded the British presence in the area to the Moluccas, Java, and other Dutch possessions in the East Indies during the Napoleonic Wars. He was then created Viscount Melgund and Earl of Minto. He died at Stevenage on 21 June 1814 and was buried in Westminster Abbey. His sister was the wife of William Eden, 1st Baron Auckland. His father's cousin Elinor (1724-1797) married John Rutherfurd (1712-1758) of Edgerston. Elinor and John's son John (1748-1837) was married to Mary Anne Leslie-Melville daughter of Major General Alexander Leslie-Melville. A brother of John Rutherfurd of Edgarston was Walter Rutherfurd (1723-1804) who married Mary sister of Lord Stirling and who were the parents of Senator John Rutherfurd(1760-1840). A nephew of Walter and John named John Rutherfurd (1746-1830) who was captured during Pontiac's Rebellion of 1763; escaped and became a Captain in the Black Watch and who settled in Mossburnford.
The Earl's second son was Admiral Sir George Elliot (1784-1863), who as a youth was present at the Battle of Cape St Vincent and the Battle of the Nile, and who was Secretary to the Admiralty from 1830 to 1834. A nephew of the Earl was Sir Charles Elliot (1801-1875), also an admiral, who took a prominent part in the war with China in 1840. Afterwards he was governor of Bermuda, of Trinidad and of St Helena.