Spurs entered Sunday's fixture following a lacklustre display in a midweek first-leg 1-0 defeat in the Netherlands and they had looked set for another loss when Marcus Tavernier gave the Cherries a deserved half-time lead.
Cristian Romero's return after a three-month absence only contributed towards an anxiety-ridden atmosphere as he misplaced early passes and, when Evanilson doubled Bournemouth's advantage after 65 minutes, the ground threatened to turn toxic.
But Pape Sarr's cross-cum-shot provided Tottenham with a fortunate lifeline two minutes later, which they gratefully accepted before Son Heung-min won and then converted an 84th-minute penalty.
It earned Spurs a point which does little for their faltering league campaign, but lifts spirits before the visit of AZ on Thursday in the second leg of their last-16 clash.
"Chaotic game, for sure. We added to the chaos and it was a little bit self-inflicted, especially in the first half. Just with the ball, we were really careless with our passing," Postecoglou said.
"At 2-0 down, it would have been very easy, particularly with the atmosphere as it was, for the boys to just let the game get away from us, but credit to them, they found a way to get back into the game and we got something out of it.
"We just kept giving the ball away. We certainly need to be better in the way that we execute that part of our game (for Thursday).
"You can imagine the noise around if the players didn't respond. Thursday is obviously a massive game for us.
"At least from that perspective - and even for the players themselves - we've obviously got a challenge on Thursday night but they overcame a fairly big challenge today and hopefully we show the same kind of mindset."
Micky van de Ven returned for his first league fixture in three months and lifted a noticeably nervous crowd, which Postecoglou conceded may well respond positivity on Thursday if his team can provide a strong start in a win-or-bust fixture with AZ.
"I can't dictate to people how they interact, but AZ's crowd certainly helped them on Thursday night and we need our crowd to get behind the boys on Thursday to make it an atmosphere where maybe we lose some of that nervousness and anxiety within our game.
"The reality is, we as a team have to respond irrespective of the atmosphere and one may follow the other. If we start the game strong, it'll get the place nice and loud, which will help us."
