Drew was born in 1904 into an African-American middle-class family in Washington, D.C. His father, Richard, was a carpet layer and his mother, Nora Burrell, was a Teacher. Drew and his siblings grew up in D.C.'s Foggy Bottom neighborhood and he graduated from Dunbar High School in 1922. Drew won an athletics scholarship to Amherst College in Massachusetts, where he graduated in 1926. An outstanding athlete at Amherst, Drew also joined Omega Psi Phi fraternity. He attended medical school at McGill University in Montreal, Quebec, Canada, receiving his MDCM in 1933, and ranked 2nd in his class of 127 students. A few years later, Drew did graduate work at Columbia University, where he earned his Doctor of Medical Science degree, becoming the first African American to do so.

Beginning in 1939, Drew traveled to Tuskegee, Alabama to attend the annual free clinic at the John A. Andrew Memorial Hospital. For the 1950 Tuskegee clinic, Drew drove along with three other black Physicians. Drew was driving around 8 a.m. on April 1. Still fatigued from spending the night before in the operating theater, he lost control of the vehicle. After careening into a field, the car somersaulted three times. The three other Physicians suffered minor injuries. Drew was trapped with serious wounds; his foot had become wedged beneath the brake pedal. When reached by emergency technicians, he was in shock and barely alive due to severe leg injuries.

In late 1940, before the U.S. entered World War II and just after earning his doctorate, Drew was recruited by John Scudder to help set up and administer an early prototype program for blood storage and preservation. He was to collect, test, and transport large quantities of blood plasma for distribution in the United Kingdom. Drew went to New York City as the medical Director of the United States' Blood for Britain project. The Blood for Britain project was a project to aid British Soldiers and civilians by giving U.S. blood to the United Kingdom.

In 1941, Drew's distinction in his profession was recognized when he became the first African-American surgeon selected to serve as an examiner on the American Board of Surgery. Drew had a lengthy research and teaching career and became a chief surgeon.

Drew was taken to Alamance General Hospital in Burlington, North Carolina. He was pronounced dead a half hour after he first received medical attention. Drew's funeral was held on April 5, 1950, at the Nineteenth Street Baptist Church in Washington, D.C.