Webber was educated at Dummer Academy (now known as The Governor's Academy) and Harvard College (B.A., 1784; M.A., 1787) where he distinguished himself in mathematics. He was ordained as Congregational minister in 1787 and two years later became Hollis Professor of Mathematick and Natural Philosophy at Harvard. He served in the commission that drew the boundaries, later recognized by the Treaty of Paris, between the new United States of America and the surrounding British provinces. He served as vice-president of the American Academy of Arts and Sciences and authored System of Mathematics, which for many years served as the only text-book on the subject in New England.
Webber was appointed president of Harvard in 1806. That same year he received an honorary Doctor of Divinity degree from that institution. He led Harvard until his death in 1810.