Vicario, 26, and Udogie, 20, arrived at Spurs from Serie A in the summer, the latter after spending a season on loan back at Udinese, and have started all six matches in Postecoglou's unbeaten League start.
Along with Micky van de Ven, January signing Pedro Porro and Pape Sarr, who was peripheral last season, the Italian pair have looked right at home in the Australian's new-look team, despite their lack of experience at this level.
Postecoglou says key to their fast transitions is the freedom to play without fear of failure.
“I try to give them as much security as I possibly can, for them to know that there's nothing that's going to happen out there that's going to be terminal in terms of their careers,” Postecoglou said.
“[My approach is]: ‘Don't stress about the mistakes, don't stress about not playing well. I'll back you, as long as I'm seeing that you're improving and working hard and doing all the things I need you to do’.
“Because part of that stress as a young player is feeling like you need to play really well to keep your place in the team and fearing that if you make a mistake in front of the whole world that's going to cost you something.”
Postecoglou continued: “It's not how I think. It's not how I'm wired, I don't care about that. This is what we're doing.
“We're going to go down this road and be successful and I try to provide that framework so at least it releases that anxiety, and they can play as themselves.
“And then it's up them. There's no guarantee. If their performances drop, then of course they won't play but I think it helps the young players focus on being as much themselves without stressing too much about outcomes.”
Vicario joined Spurs from Empoli in a £17.2million deal, having worked his way up from the fourth-tier of Italian football.
“You get presented with some options and I just really liked Vic from the start in terms of the characteristics he has as a goalkeeper," Postecoglou said. “His agility, his demeanour, his character.
“Then I spoke to him and he's just such an infectious personality. He was so determined he wanted to come to Tottenham and prove himself at the next level.
“He hasn't had a long time at the top level, similar to Micky van de Ven. I like players who have worked their way up from a low level very quickly because it shows that they adapt very quickly but also that they're coachable, that they learn.
“And you look at Vic, and he's taken big steps very, very quickly and he's been absolutely outstanding for us in the first six or seven games. He's been a really strong influence on the field in the big moments when we've needed him. And he comes in every day and wants to improve, and that's exactly what we need.”
Udogie has been a revelation as Postecoglou's inverted left full-back and last weekend caught the eye in an intense duel with Bukayo Saka in Spurs' 2-2 draw with Arsenal.
On Saturday, he is likely to be influential again when Spurs host Liverpool and Mohamed Salah.
“That's the Premier League, that's why he's here," Postecoglou said. “There's no greater learning curve for him.
“You saw last week, he had to work really hard against a top opponent but in the end he came out pretty good and he can't rest on that because he's got Salah this week and then he'll have somebody else, a [Jack] Grealish or someone else, coming in the next weeks and he's got to cope with all of it, but so far he's doing very well.
“When I got the role I had a look at all the players who were out on loan last year and Destiny was one. I really liked his profile. Physically he's got outstanding attributes but he obviously played a different style of football in terms of a lot of times he was used as an attacking wing-back.
“He made a real impact in a tough league as a young man. Serie A is a tough league. He handled himself really well and it was one of those where as soon as we got him in I could see that he wants to improve every day.
“He's got outstanding attributes and for a 20-year-old he's already had some really big challenges, defensively and he's handled himself so well.
“When he has the ball and he runs forward he looks exciting. Really pleased to have him in the group.”