Glaze was born in Denver, Colorado, and was recruited by Dartmouth College after displaying his skill in two sports. He played football at the University of Colorado in the 1901 season under coach Fred Folsom, a Dartmouth alumnus who became that school's coach in 1903. Glaze enrolled at Dartmouth in 1902, being followed there by his younger brother, John. Under Folsom, he played a notable role in the school's first-ever football victory over Harvard in 1903, a game in which Harvard dedicated its new stadium. In 1905, Glaze was named an All-American as an end by Walter Camp, even though at 5'8" and 153 pounds he was the smallest player on Dartmouth's team that year. Glaze also played baseball at Dartmouth, and pitched a no-hitter against Columbia.