Twilley was runner-up for the Heisman Trophy during his time at Tulsa before he was picked by the Dolphins in the 12th round of the 1966 AFL draft. While he was also selected in 14th round of the NFL draft by the Minnesota Vikings, Twilley chose Miami where he spent his entire 11-year professional career.

A year after the Dolphins were hammered 24-3 by the Dallas Cowboys in Super Bowl 6, Twilley scored the franchise’s first Super Bowl touchdown, a 28-yarder from Bob Griese that gave the Dolphins a 7-0 lead on Washington en route to closing out the 17-0 perfect season.