The 24-year-old England international was received warmly by the home fans at the end of a week during which he asked for a move from his home-town club, and responded with a fine individual display.
It was Downing's pass which set up record signing Afonso Alves to fire his side into the lead in the final minute of the first half, although his overall performance was just what his manager had been looking for.
Southgate said: "He was excellent. There were two things about it, really: one, he got a brilliant response from the supporters before the game, which will have been a huge lift for him, and then he gave a performance full of quality, but also endeavour.
"He worked his socks off for the team, got back and won tackles, got back to intercept things.
"He is a class player and he showed today why it is so important for us to hang on to him."
Southgate insisted Downing did not actually hand in a written transfer request, but admitted he had asked about a move as Tottenham formalised their interest in him.
Having been told he would not be sold, he has committed himself to the club for the rest of the season, although what happens then remains to be seen.
Southgate said: "We have just said, there is nothing going to happen now, and what happens in the future happens in the future.
"None of us can look that far forward. We have got a job to do here.
"We have got on with it. He understands our situation. As I said earlier in the week, he loves playing for this football club and nothing would give him greater pleasure than for him to be playing here and us being up at the top of the league.
"Now that's obviously very difficult, but he knows his job between now and the end of the season is to help this club get as high as we possibly can.
"He has been in and he has asked about the possibility of a move. He knows we have had bids from Tottenham because that's the way they work.
"We have had to deal with it the majority of the week and it has been an unbelievably long week, so I am glad we come out with something from it."