Moegle attended
Taylor High School, where he played football and basketball. He accepted a football scholarship from
Rice University and attended as a 16-year-old freshman. His play was limited in his sophomore season (1952), after he was lost with a hand cut he suffered trying to open a classroom window that was stuck. In 1953, he teamed up with fullback
Dave "Kosse" Johnson (the nation's second leading rusher), to win a share of the
Southwest Conference title with the
University of Texas, while registering 833 rushing yards with a 7.3-yard average, which led the nation. In the
1954 Cotton Bowl Classic against
Alabama, Moegle was involved in one of college football's most famous plays. With
Rice leading 7–6, Moegle broke through on a
sweep from Rice's five-yard line, and was running down the sideline in front of Alabama's bench on his way to a touchdown. Alabama's
Tommy Lewis, without putting on his helmet, jumped off the bench and tackled Moegle. Seeing what happened, referee Cliff Shaw awarded a 95-yard touchdown on the play, and Rice went on to win the game 28–6. Moegle finished with 265 rushing yards, which was a Cotton Bowl Classic record until the
2008 game when
Missouri's
Tony Temple rushed for 281 yards. Moegle and Lewis later appeared on
The Ed Sullivan Show to talk about the play. As a senior in 1954, he rushed 905 yards and led the nation in punt returns. He finished his college career after setting 26 school records, including career touchdowns (22), interceptions in a game (3) and total points in a season (72). In 1979, he was inducted into the
College Football Hall of Fame. In 1970, he was inducted into the Rice Athletic Hall of Fame. In 1980, he was inducted into the
Texas Sports Hall of Fame. ==Professional career==