Maddison pulled the strings on the south coast and opened his account for Spurs with his 17th-minute goal, which helped make it back-to-back victories in the Premier League for the visitors.
Dejan Kulusevski added a second after half-time to end his goal drought stretching back to January to keep the feelgood factor around Tottenham despite losing record marksman Harry Kane to Bayern Munich on the eve of the new campaign.
Postecoglou has overseen an impressive seven-point haul from a possible nine since he took over and admitted he could now enjoy his 58th birthday on Sunday.
“Yeah, I’m 58. We always put birthday celebrations on hold depending on the result but I’m sure my wife is scrambling to arrange something for tomorrow,” the Australian smiled.
“It’ll be nice. Short turnaround to Fulham so still got to go into work tomorrow but it’ll be nice to spend some time with the family and now another click closer to that 60 mark, mate.”
There had been doubts over Maddison’s availability at Vitality Stadium after he left Tottenham Hotspur Stadium last weekend on crutches and in a protective boot.
While the England international said at the time it was precautionary, he only returned to training on Friday, but starred again with the number 10 shirt on his back.
Maddison moved from Leicester in June for an initial £40million deal that could rise in add-ons and during a summer where midfielders have transferred for more than £100million, it appears a steal.
Postecoglou added: “A £45 million bargain? I don’t know what world you live in mate, but I know what you mean and relatively so.
“There wasn’t anyone happier than me when we got him. I was delighted. I wouldn’t say I’m surprised, but I’m overjoyed at the footballer I’ve got. The way he’s embraced the whole club, where he’s at in his life.
“He really wants to be the person. He’s that creative force for us. He works hard, he wins the ball back, presses. It’s not like he just comes to life when we’ve got the ball. He’s had a disrupted week. He only had one session with us, but for him to put on a performance like that is credit to him.
“And I think there’s more to come when he gets more understanding with our forward players and they get more understanding with him. He’s going to keep improving. If we got him in the bargain bin, that’s great for us.”
The only negative for Tottenham was another game without a goal for Richarlison, who was replaced on the hour mark.
“He just needs to work hard, keep contributing to the team and contributing to us being successful and overcoming the challenges we have,” Postecoglou countered.
“Richy will work hard. He’s a good footballer and he’ll get his goals.”
Meanwhile, Bournemouth boss Andoni Iraola was left disappointed with their end to an entertaining contest.
Ryan Christie tested Guglielmo Vicario before half-time and Antoine Semenyo curled into the side-netting after the break before Destiny Udogie burst into the area and cut back for Kulusevski to wrap the points up for Spurs.
Iraola said: “I was disappointed probably with the end of the game after the 2-0.
“I think until 2-0 the team was playing really well and it looked like in this moment we had chances to score the 1-1, but from there we probably lacked a little bit of intensity and they were more comfortable on the ball.
“In the first half we put them in really difficult positions. I think (Yves) Bissouma and Maddison were in difficult positions, but they are good and they were winning duals even with a man on their back and protecting very well the ball.
“Probably we didn’t regain so many balls because for me they could sustain a very good press.”