Rashford scored the only goal of the game at the King Power Stadium as England ended a run of three successive defeats with their first victory since the quarter-finals of the World Cup.
The 20-year-old has now scored in back-to-back international games for the first time, and Southgate believes that the Manchester United striker has a bright future ahead of him.
"I am really pleased for Marcus Rashford. We know how exciting he is, he will be a top player," Southgate is quoted as saying by BBC Sport.
"He has a super mentality, some of his performances have been really good but to finish with two goals is good for him."
Rashford has now scored his five England goals at four different grounds in the country.