Davis was born and raised in Southern California, the son of a bank manager. Glenn and his twin brother Ralph played high school football at Bonita High School in La Verne, California. In 1942, Davis led the Bearcats to an 11-0 record and the school's first-ever football championship, earning the Southern Section Player of the Year award. In 1989, Bonita High's stadium was dedicated in his name. The brothers were close and had originally planned to attend USC in Los Angeles, but when their U.S. Representative agreed to sponsor them with appointments to West Point they decided to go there.
At West Point, under coach Earl Blaik, Davis played fullback in his freshman season. Blaik moved him to halfback for his three varsity seasons, while Doc Blanchard took over at fullback. With Davis and Blanchard, Army went 27-0-1 in 1944, 1945, and 1946. Davis was nicknamed "Mr. Outside", while Blanchard was "Mr. Inside".
Davis averaged 8.3 yards per carry over his career and an astounding 11.5 yards per carry in 1945; both results are records which still stand today. Davis led the nation in 1944 with 120 points. He scored 59 touchdowns, including eight on his freshman squad, in his career. His single-season mark of 20 touchdowns stood as a record for 10 years.
He and Blanchard set a then-record 97 career touchdowns by a pair of teammates. (The record was broken by USC backs Reggie Bush and LenDale White, who had 99 career touchdowns.) In 2007, Davis was ranked #13 on ESPN's list of Top 25 Players In College Football History.
For all three varsity years at West Point, Davis was a "consensus" All-America player (that is, selected by all the different groups picking All-America teams). In 1944, he won the Maxwell Award and the Walter Camp Trophy, and was runner-up for the Heisman Trophy. In 1945 he was again runner-up for the Heisman (won by his teammate Blanchard). In 1946, he won the Heisman and was named the Associated Press Male Athlete of the Year. In 1961, Davis was inducted into the College Football Hall of Fame.
Davis graduated from West Point in June 1947 and entered the U.S. Army as a second lieutenant. He was offered a contract and $75,000 signing bonus by the Brooklyn Dodgers, but declined, as he was required to serve in the Army and would be a relatively old rookie after that.
Davis was also denied extended furloughs or other accommodations that might allow him to play football while serving in the Army. There was public feeling that after the expense of his West Point education, he should not just go off to play football.
Davis (and Blanchard) did earn $25,000 each by appearing in the low-budget movie Spirit of West Point (about their football careers). Davis tore a ligament in his right knee during filming.
Davis served three years in the Army. While on leave in 1948, he attended the Rams training camp and played in a pre-season game. He then reported for duty in Korea (this was before the Korean War, which began after he returned to the U.S.).
Despite his knee injury, Davis was an effective player, and was even named to the 1950 Pro Bowl. But in 1951, he injured his knee again. He was out for the 1952 season.In September 1953, the Rams released him, ending his professional career.