Tahith Chong gave Luton a surprise early lead before Tottenham equalised in the 51st minute through an Issa Kabore own goal.
Son then finished off a flowing team move in the 86th minute to secure a 2-1 win with his 160th goal for the club, which put him fifth on their all-time scorers' list.
"I think the thing about Sonny is that irrespective of opposition or whatever game it is, he has this really high level of expectation on himself to be the best he can be all the time. It's very hard to be like that," Postecoglou told reporters.
"It's a lot easier to try and get comfortable in the state of being as a person or as a player because he's done so much already. But he wants to maintain the highest possible standards whenever he plays, and when you see that with his football, his efforts, his leadership now so it's all encompassing.
"He's been a brilliant footballer for this club for a long time and hopefully for many years to come, and again I thought he was really unlucky with a couple of chances and some good saves. But when we needed him he was there and got the job done."
Tottenham, fifth in the league standings, next travel to face seventh-placed West Ham United in a league match on Tuesday.
