Spurs fought back from 2-0 down to draw at home, with Heung-min Son converting a late penalty.
Postecoglou said afterwards: "At 2-0 we had a bit of a mountain to climb. It was a chaotic game today and the boys showed a real mentality. It would have been really easy at 2-0, with the atmosphere in the stadium, to let it slip away from us but they fought hard and got themselves back in the game.
"It was a different kind of struggle for us today. Bournemouth pressed really well and we were really sloppy with the ball at times which allowed them to pin us back. It wasn't anything like Thursday but we hung in there.
"There is definitely a sense of nervousness there. We are getting players back who have been out quite a while. We've been keen to get Romero back and his first pass shows he's not played for three months. Some of it is just anxiety and nervousness which we need to get under control.
"I think Micky van de Ven, Dom and Romero are quality players and we've hardly had them all year. It gives the team a bit of a boost to see them out there. It would have been a fairly deflating feeling had we not. It gives us and the players some understanding that when we stay strong and we can overcome whatever kind of struggle we have.
"I don't preach from the altar. People are allowed to feel the way they do. Playing away at AZ they certainly helped their team and we'd like to think on Thursday we will make it just as hostile for the opposition."