McClaren was in charge of Southgate, who played as a defender, during his five years at Middlesbrough, when they won the League Cup in 2004 and reached the UEFA Cup final.
The Derby boss told Sky Sports News: "Gareth has made the progression. He's got the experience, international football is totally different from club football, and Gareth has had more experience with the U21s, qualifying for tournaments and dealing with tournaments. That will be invaluable for the senior team.
"The big test will be in couple of years at the World Cup finals. I wish him luck and I know him very well. He was always a student of the game and gets on well with everybody. He speaks to the players on their level. He had prepared for many years for this and deserves to get this job.
"I made him captain. He led the dressing room, was always a leader. He's got all the credentials you would want and he's got a big opportunity now."
Southgate's first official game as England manager will be a friendly with Germany on March 22.