Stephenson was an All-American at the University of Alabama, under coach Bear Bryant. Bryant called Stephenson the best player he ever coached, regardless of position. He was drafted by Don Shula and the Dolphins in the second round of the 1980 draft. The 6'2, 255 pound center would eventually become regarded by many as the best center of all time, but was used on special teams only until late in the 1981 season, when Mark Dennard was injured.
With the exceptionally explosive Stephenson as offensive captain, the Dolphins offensive line gave up the least sacks in the National Football League for a record 6 straight seasons, from 1982-1987, which doubled the length of the previous record. After he left, the Dolphins' line would continue to protect the pocket superbly, extending the record to 9 straight seasons.
Stephenson was voted as an All-Pro five consecutive times from 1983-1987. He was selected to play on five Pro Bowl squads over the same span. He started at center in the AFC Championship Game three times, in 1982, 1984, and 1985. He was the starting center in the last two Dolphin Super Bowl appearances:Super Bowl XIX and Super Bowl XVII. In 1985, Dwight was the recipient of the Walter Payton Man of the Year Award. In 2005 he was named the Walter Camp Man of the Year.
Despite the brevity of his career, in 1999, he was ranked number 84 on The Sporting News' list of the 100 Greatest Football Players. Also, in 1999, he was inducted into the Virginia Sports Hall of Fame.
His son, Dwight Jr. was a linebacker and defensive lineman at the University of Notre Dame from 2003-2007.