Spurs won 5-0 on Sunday night, with the result leading to the dismissal of Saints manager Russell Martin.
Postecoglou said: "Really proud of the players. We obviously had a tough away European fixture on Thursday night. We had 10 first-team players unavailable for a number of reasons today in a squad of 25, and we asked a lot of them to go out there again and dig into their wells of energy.
"A couple we put out there hadn’t played in ages, haven't started games, so the fact the boys could play with such energy and quality was just outstanding. Really pleased that they get the rewards for it because they deserve it."
On James Maddison and Heung-min Son, he said: "They were brilliant. I think they sensed they needed to tonight. We are just spread so thin that we’re asking young players and players who haven't played a lot and others who have been playing all the time to go out there and still try to play to our identity, play the football we want to.
"I thought they both were the catalyst for that tonight, with both the football but also their mindset just going out there. We had to start strong today, we knew we would probably run out of gas at the end of the game considering the line-up we had out there, coming off Europe on Thursday night. I think the players sensed that and credit to them that they started the game in that way."