He was born the son of Elizur Goodrich of Durham, studied at Yale University until 1779 Law and was admitted to the bar in 1783. After working for a time as a lawyer in New Haven, he was from 1795 to 1802 Member of the House of Connecticut.
In the presidential elections in 1796, Goodrich was an elector of the Federalist Party. He was elected for Connecticut House of Representatives of the United States, however, where he was active only from 1799 to 1801. Then he was appointed by John Adams to the customs inspector of the port of New Haven.
Goodrich also taught at Yale University, worked from 1802 to 1818 on probate and 1803-1822 mayor of New Haven. He died in 1849 and was buried in the Grove Street Cemetery in New Haven. His older brother, Chauncey was also a congressman and U.S. senator for Connecticut.