They defeated the Oakland Raiders in the AFL Championship game and faced the Baltimore Colts of the National Football League in Super Bowl III.
With the Jets installed as an 18-point underdog, Beverly's small stature made him a main target of Colts' quarterback Earl Morrall.
Driving the Colts down the field and poised to score the team's first touchdown of the game, Unitas threw a crossing pattern to one of his wide receiver's in the end zone, but Beverly stepped in front and caught the ball in the end zone, downing it for a touchback.
While the Colts managed to score a late touchdown, the Jets' 16–7 triumph is considered to be one of the greatest upsets in the history of professional sports in the United States.
He later played in the World Football League with former Jet Super Bowl alumni Gerry Philbin, George Sauer, Jr., John Dockery, and Vito (Babe) Parilli with the New York Stars.
Grand children's names are from oldest to youngest; Sean, Vanessa, Briana, Victoria, Brittany, Stefania, Fabio Randy, Isaiah Pierce and Tove.
One of his sons, Randy Beverly Jr., was the head coach of an American football team in Sweden, Örebro Black Knights.