Minto was born in Edinburgh, the eldest son of Sir Gilbert Elliot, 3rd Baronet, and Agnes, daughter of Hugh Dalrymple-Murray-Kynynmound.

About 1763 Elliot and his brother Hugh were sent to Paris, where their studies were supervised by the Scottish philosopher David Hume, and where they became intimate with Honoré Mirabeau.

He became very friendly with Edmund Burke, whom he helped in the attack on Warren Hastings and Sir Elijah Impey, and was twice an unsuccessful candidate for the office of Speaker,[3] in the elections of January 1789 and June 1789.

From 1799 to 1801 he was Envoy-Extraordinary to Austria, 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,[3] with his term starting on 31 July 1807.

He governed 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.