[3][4] While he was in high school, he swam for the Wilmington Athletic Club under Coach Bob Mattson, along with his younger brother David Johnson.

Johnson's brother Dave was a U.S. Olympian who competed in the preliminaries of the 4x100 and 4x200 freestyle relays in Mexico City in 1968, but did not medal as he did not swim in the finals.

[2][5] Johnson attended Salesianum School, an initially all-male independent Catholic institution in Wilmington, with a strong sports program.

[1] He began his studies at University of Delaware in the fall of 1962, and swam for Coach Harry Rawstrom from around 1962-66, setting a total of nine school records, including marks in the 100 and 200-yard butterfly events.

[8] In 1979, he became head coach of the women's swim team at Delaware amassing an impressive 58–4 record, which included a string of 42 consecutive wins between 1979 and 1982.

[2][17] His women's swim team at Delaware twice won the EAIAW Division II Eastern Championships, the 80-81, and 81-82 seasons.

One of his top swimmers, Mary Pat Johnson, had been a member of Delaware's 200-meter freestyle relay team and had noted that by February 1982 the team had had two undefeated seasons, and Johnson had established a remarkable 38-1 dual meet record with their only loss against Pennsylvania's Shippensburg University.

During his time in the position, he oversaw construction of a new men's and women's basketball venue, known as the Bob Carpenter Center.

[16] In his honor, The Edgar Johnson Award is given each year to the University of Delaware senior male athlete who best exhibits the characteristics of dedication, hard work, and fairness, and consistently reaches for excellence.