Weaver received three degrees from the University of Wisconsin–Madison: a Bachelor of Science in 1916, a civil engineering degree in 1917, and a Ph.D. in 1921. He became an assistant professor of mathematics at Throop College (now California Institute of Technology). He served as a second lieutenant in the Air Service during World War I. After the war, he returned to teach mathematics at Wisconsin (1920–32). Weaver married Mary Hemenway, one of his fellow students at Wisconsin, a few years after their graduation. They had a son, Warren Jr., and a daughter, Helen.