Raul Jimenez's superb solo goal separated the sides as Wolves picked up their second win of the season.
And while Lage insists the result is for the fans, he also praised his side for the way they exploited the space against the Saints.
"We tried to play the way we want to play," Lage said.
"The changes I did to the team were to have more of the ball. We saw what Southampton did against Man City and Newcastle and they are a good team. What Ralph (Hasenhuttl) did in his three years here is very good.
"When you play against this team, they have the ball and come with a big intensity to press high. We tried to do little things different and get Joao and Leander to find spaces in between the lines. That's why Daniel and Hwang started, to have more of the ball.
"The first 45 minutes we managed the ball and were solid and after found the right spaces to create our chances. We started with a good chance from Semedo.
"The plan was to keep the ball and wait for our solutions. We need to understand the spaces the opponent gives. The team were clever to understand that and which spaces to go to attack.
"It was a good result for us, three good points. We are happy with that.
"These three points go straight to our fans because the way they supported us from the first minute was very important to stay the 90 minutes with personality.
"They are a hard team to play against and it's a very hard stadium to come and win the three points."
Lage made some surprising and brave decisions with his team selection by dropping Ruben Neves, Francisco Trincao and Adama Traore – as well as not bringing Willy Boly back into the defence.
The head coach recently said no player is undroppable and he has no 'personal relationships' with the squad so he can make those tough decisions – which he says were purely tactical against Southampton.
When asked if it was a tactical decision, Lage added: "Yes. Our job is to take decisions. It's hard to take decisions.
"In the end when you win, it doesn't mean you were good or bad in your decisions. We had in mind to put Joao more in the middle to come behind their strikers, because they come with everything to press.
"Joao, for me, is our best midfielder to find spaces in between the lines and spaces behind the midfielders.
"We put Daniel and Hwang there to have that space and have the ball.
"I don't know if it was brave or not. When I look to the bench I see strong players like Ruben, Adama, Trincao – they did well.
"Also Boly, Fabio, Ki – I don't want to forget anyone. When you look to the bench you can see we have a good squad to change the game.
"We started with these guys and after Ruben was important to come into the game and help us be more solid. Adama is important because the spaces were different to play and also Ki came after a good performance against Tottenham to help us to defend the last five minutes."