Sancho has impressed for Borussia Dortmund this season, contributing more assists than anyone else in Europe, leading to his first call up to the England senior squad.
Gareth Southgate handed him his debut off the bench against Croatia on Friday night, and Smith believes this is only the beginning for the Londoner.
Smith said: "He is so swift and skillful. He had the confidence to take on his man, nice subtly on his cross that caused a lot of problems.
"He looked like he just went out there and enjoyed it. It is not an easy way to make your debut, but he did grab it with both hands.
"I think people rightfully should be excited. Let's not get ahead of ourselves, but he is a young lad with talent, confidence and maybe an old head on young shoulders to handle the situation for England."