Leading by two goals with 25 minutes to play, and the better side from the start, Andoni Iraola said it was “a missed opportunity” for his men, while Postecoglou can reflect on his positive second-half substitutions sparking a Spurs resurgence.
After Marcus Tavernier and Evanilson had put the Cherries in charge, Pape Matar Sarr and a penalty from Heung Min Son brought Tottenham back to get a draw that eases the pressure on Postecoglou.
“It was chaotic, and we added to the chaos,” he said. “We looked nervous in the first half, but showed a strong mindset to come back, and hopefully can show the same on Thursday.”
The biggest surprise was that it took Bournemouth 42 minutes to score, so poor were Spurs in the opening half. Cristian Romero's return from a three-month injury layoff was greeted with a roar by Spurs supporters, but within minutes it was clear that the Argentine was ring-rusty.
Only 20 seconds had elapsed before he passed sideways to Evanilson, whose low shot was well saved by Guglielmo Vicario.
Ten minutes later Romero gave the ball away again, and when Rodrigo Bentancur did the same, Vicario again saved well to deny Justin Kluivert.
Spurs were a shambles in defence, and their problems were primarily of their own making, losing possession time and time again. Tavernier's goal, three minutes before half-time, was a case in point.
Pedro Porro led a break into the Bournemouth half, but his tame pass towards Brennan Johnson was easily cut out by Milos Kerkez, who charged down the left flank before delivering a delicious, curling cross to the far post that Tavernier converted with a sliding finish.
It was exactly what Bournemouth deserved for their bright, smart football, with Iraola showing why he he is being talked about as a possible successor to Postecoglou.
The Spurs manager had opted for a workmanlike midfield, dropping James Maddison and Lucas Bergvall along with captain Son, but it meant Spurs carried little attacking threat.
Spurs were booed off at half-time, and Postecoglou responded by sending on Bergvall and Son, who went close with a shot that was tipped round the far post. Justin Kluivert had a goal ruled out for offside after a VAR review, before setting up Evanilson to make it 2-0 with a delightful clipped finish in the 65th minute, lifting his shot over Vicario.
That should have been it, but Spurs showed there is spirit in the side, despite their travails. Sarr pulled one goal back a minute later with what appeared to be a mis-hit cross from the right that flew in off Kepa's far post.
Maddison, another substitute, then put Son in, and when Kepa sent the Korean flying, referee John Brooks pointed to the penalty spot.
Son sent him the wrong way to snatch a draw Spurs barely deserved.
Tottenham (4-3-3): Vicario 7; Porro 5, Romero 4 (Van de Ven 61), Danso 6, Spence 7; Sarr 6, Bissouma 5 (Bergvall 46), Bentancur 5 (Maddison 61); Johnson 5 (Son 46), Solanke 5, Odobert 6.
Bournemouth (4-4-2): Kepa 6; Cook 7, Hill 7, Huijsen 7, Kerkez 8 Tavernier 7 (Brooks 81), Christie 6 (Scott 71), Adams 6, Kluivert 7; Evanilson 7 (Ouattara 71), Semenyo 6 Sinisterra 89).
Referee: John Brooks 7