Robinson was born in Hardwick, Massachusetts where he spent his childhood. As a young man he pursued classical studies. In 1761 he moved with his family to Bennington, Vermont. He soon became an important citizen of Bennington, serving as town clerk from 1762 to 1781. Meanwhile, he studied law and became active in the American independence movement, serving as a colonel in the Vermont militia during the early parts of the Revolutionary War. In 1778, when Vermont became an independent republic, Robinson became a member of the government council and the chief justice of the Vermont Supreme Court. In 1782 he was sent to the Continental Congress as a state agent to solve a boundary dispute with New York. He served on the government council until 1785 and as chief justice until 1789, when he became governor of Vermont, replacing Thomas Chittenden. Robinson served as governor until 1790 shortly before Vermont was admitted as a state to the United States.
Robinson was then elected by the Vermont General Assembly to one of Vermont's two United States Senate seats. He served in the Senate for one term, from 1791 to 1797. He became associated with the anti-administration faction and, later in his term, with the beginnings of the Democratic Republican Party of Thomas Jefferson. After his retirement from the Senate, Robinson moved back to Bennington and practiced law. He served in the Vermont State House of Representatives in 1802. He died in Bennington, and is interred in the Old Bennington Cemetery.
Robinson is also well known for receiving a letter from Thomas Jefferson in 1801 in which Jefferson said that if Christianity were simplified, it would be a religion friendly to liberty. Moses Robinson was the older brother of Jonathan Robinson, was also prominent in Vermont's political history.