Born in Lyme, New London County, Connecticut to Matthew Griswold and Ursula (Wolcott) Griswold; pursued classical studies, and was graduated from Yale College in 1780; studied law; was admitted to the bar in 1783 and commenced practice in Norwich, Connecticut; returned to Lyme in 1794; elected as a Federalist to the Fourth and to the five succeeding Congresses and served from March 4, 1795, until his resignation in 1805 before the convening of the Ninth Congress; chairman, Committee on Revisal and Unfinished Business (Sixth Congress), Committee on Ways and Means (Sixth Congress); declined the portfolio of Secretary of War tendered by President John Adams in 1801; served as a judge of the supreme court of Connecticut in 1807; presidential elector on the Charles Cotesworth Pinckney and Rufus King ticket; Lieutenant Governor of Connecticut 1809-1811; Governor of Connecticut from 1811 until his death in Norwich; interment in Griswold Cemetery at Black Hall, in the town of Lyme (now Old Lyme, Connecticut).

Griswold was grandfather of congressman Matthew Griswold. Griswold's father (Matthew Griswold), his maternal grandfather (Roger Wolcott), his uncle (Oliver Wolcott), and his cousin (Oliver Wolcott, Jr.) were all also Governors of Connecticut.