Maddison opened the scoring in the 17th minute, slotting the ball into the bottom-left corner after latching on to a sublime through-ball from Pape Matar Sarr.
In scoring at Vitality Stadium to add to his two assists against Brentford in Matchweek 1, Maddison has now been involved in 18 Premier League goals in 20 away matches.
Maddison's outstanding performance came after he left the match against Manchester United a week ago with a protective boot on.
After half-time, the hosts had a period of sustained pressure, but Antoine Semenyo's long-range effort hit the side of the net and Dominic Solanke put in a dangerous ball across the box, only for no team-mates to be in a position to score.
Spurs made the three points safe when Dejan Kulusevski scored his first Premier League in 20 matches on 63 minutes. Destiny Udogie cut the ball back into the six-yard area for the Swede to finish from close range.
The victory for Postecoglou's team took them temporarily to the top of the table ahead of the 15:00 Saturday matches, although they were overtaken at the end of the day by West Ham United on goals scored after their 3-1 win at Brighton & Hove Albion.
This was only the third time Spurs have scored at least twice in each of their first three matches of a Premier League season. They have also kept successive clean sheets under the Australian.
'Excellent first half'
Postecoglou was especially pleased with how his team coped with Bournemouth's pressing game in the opening 45 minutes.
“I thought the first half was excellent,” Ange told SPURSPLAY. “We knew, particularly here, that Bournemouth would be really aggressive with their press, and I really liked the way we handled that.
"We stayed clear-headed, brave when we needed to be and that helped us take control of the game, we scored a great goal and probably could have had one or two more.
“Breaking the press was the only way we’d be able to get through that, we could go long but by playing through the press it meant once we got through, there was plenty of space for us to go and attack. I really liked how we looked in those moments. There are still areas to improve, but I really liked the way the boys handled it."
Richarlison drought goes on
One negative for Spurs was Richarlison missing a couple of presentable chances to open his account this season. The Brazilian has now only one Premier League goal from 30 matches for Spurs.
Ex-Spurs striker Jermain Defoe believes Richarlison can step up and fill the hole left by the departure of Harry Kane, ideally by having studied the England striker in the time they were together at the club.
"It's up to him to hopefully get a run of games and show everyone what he can do," said Defoe. "I'd like to think that he has learnt off Harry."
Bournemouth head coach Andoni Iraola saw some positives from his side's display but felt the match was decided by individual battles in midfield.
"We gave them a game until the second goal," Iraola told TNT Sports. "From there we were worse on the press and they dictated the game.
"But we had this momentum just before 2-0. It looked like we could tie the game with the chances we had. But, from there, they were better than us.
"We were pressing really well first half. But they are very good, even with the press they can keep the ball and turn. You just need to win the duels in the middle. That we were not doing."