Furchgott was born in Charleston, South Carolina, and graduated with a BS in chemistry from the University of North Carolina in 1937. Three years later he gained his PhD in biochemistry at Northwestern University. From 1956 to 1988 he worked in the department of pharmacology at the State University of New York, and then became Distinguished Professor, State University of New York Health Science Center. In 1989 he became professor of pharmacology at the University of Miami School of Medicine, retiring in 2004. In 1998 he shared the Nobel Prize for physiology or medicine with Louis Ignarro and Ferid Murad for their discovery that molecules of the gas nitrogen monoxide (nitric oxide, NO) can transmit signals in the cardiovascular system.

Nitrogen monoxide, produced by one cell, acts by penetrating membranes and regulating the function of another cell. Nerves and hormones are well known as signal carriers, but this discovery was a totally new signaling principle in a biological system.

Furchgott began by researching the actions of drugs on blood vessels, but often observed contradictory results. Sometimes a drug acted to contract a vessel, but at other times it dilated the vessel. In 1980 he discovered that the neurotransmitter acetylcholine did not act to dilate blood vessels if the endothelium (surface cells) of the receiving cell were damaged. He deduced that the cells of the endothelium produce another hitherto unknown signal substance that makes the smooth muscle cells of the vascular system relax. He called the substance endothelium-derived relaxing factor (EDRF), and other pharmacologists (including Ignarro) set out to find it.