The Lilywhites ended a two-game losing streak in the top flight courtesy of a 3-1 success over their capital counterparts, as Dominic Solanke, Brennan Johnson and James Maddison cancelled out a 23-second Bryan Mbeumo opener.
However, Brentford felt aggrieved in the second half when Vicario appeared to handle the ball outside of the area, having made a meal of catching a delivery and taking a few swipes at the ball right on the 18-yard line.
Vicario's second flap at the ball looked borderline, although his third touch - which was deliberate - appeared to have been outside the box, leading to fierce protestations from the Bees, but referee John Brooks was unmoved.
The VAR room also did not intervene, while Brentford boss Thomas Frank and defender Kristoffer Ajer were booked for complaining before Vicario went on to make a couple of crucial saves.
Postecoglou makes tongue-in-cheek "lucky" Tottenham admission
Speaking in his post-game press conference, as quoted by football.london, Postecoglou replied to a reporter's questions with curt responses, firstly saying: "I think I saw exactly what you saw," when asked what he made of the incident.
When the reporter replied that they were not 100% sure, Postecoglou added: "Well what makes you think I am surer? I was closer to the pitch? You were higher up though. Okay, so maybe I saw exactly what you saw."
Asked if Tottenham "got away" with Vicario not seeing red for the incident, Postecoglou responded: "Get away with it? So have you seen it and said it's outside, because I haven't seen it. I've got no idea."
The journalist then claimed that the incident was not deemed to be an obvious goalscoring opportunity, hence why the VAR room did not intervene, after which Postecoglou closed the matter with a "lucky" admission.
"So, if it wasn't an obvious goalscoring opportunity, it means even if he was penalised it would have been a free-kick and that's a big turning point in the game? A free kick outside the box? (laughs) Okay, look I guess we were lucky to get the result."
Postecoglou explains late Solanke change in Brentford win
Prior to the Vicario incident, Mbeumo followed up Yoane Wissa's 22-second opener against Manchester City with an incredible 23-second volley in North London, although Solanke and Johnson - two players who have experienced troubled starts to the season - put the Lilywhites 2-1 to the good before the break.
Maddison's 85th-minute chip - from Son Heung-min's second assist of the match - then killed the game off, although Postecoglou revealed that he had hoped to withdraw Solanke a bit earlier, as his £65m striker is still not quite at full fitness.
"Dom's still working his way to match fitness. You saw at the end there, he was out of gas. We wanted to take him off a bit earlier, but we needed that third goal to kill the game off," Postecoglou said.
"I thought he led the line really well, important for us today as a link up there because we knew they'd go pretty aggressive on a man-to-man marking and he was going to have to hold the ball up for us and I thought he did that really well."
Tottenham now prepare for Thursday's Europa League opener against Qarabag before next weekend's crunch clash with Manchester United at Old Trafford.