The 19-year-old will miss the opening weeks of the season with a knee injury, but the Cherries see the England Under-20s international as a long-term investment.
Bournemouth beat Premier League rivals Wolves to his signature after he impressed last season, with even Manchester City manager Pep Guardiola hailing him as an 'unbelievable player' last season.
Scott played 49 games for Bristol City last season, scoring twice and providing five assists as the side finished 14th in the Championship.
He also won the Under-19 European Championships with his country in 2022.
Scott becomes Bournemouth's second most expensive signing in the club's history, surpassing Hamed Traore who joined from Serie A side Sassuolo for £20m, after impressive loan spell.
MORE ON BOURNEMOUTH
'Next Neymar' Jackson failed medical at Bournemouth but can solve Chelsea's scoring issues
Wolves confirm appointment of Gary O'Neil as new boss after Julen Lopetegui exit
His fee falls just short of Jefferson Lerma's price in 2018, with him signing for the Cherries for almost £30 million.
Scott has not played for the Robins so far this season due to a swollen knee, but despite his absence from the 1-1 draw with Preston and the cup win, manager Nigel Pearson admitted he's 'disappointed' to lose his star midfielder.
"I've got nothing to say, really, apart from that the club has given him permission," Pearson said, prior to deal being confirmed. "Outside of that I've got no comment. I couldn't give you the details… I knew [about the bid] today. And that's it, I'm not saying anymore."
When pushed on the topic, he added: "I'd been planning to keep him. It's nothing to do with me. It's a club decision.
Most read in Football
Bayern agree Kane fee, De Gea to Real Madrid, Liverpool hijack Caicedo, Chelsea eye Lavia
Man United release provisional squad numbers this season with Greenwood absent from list
De Gea finally set for Real Madrid transfer after dream move failed eight years ago
England star Lauren James banned for World Cup stamp but tournament not over
"I'm disappointed because we've got a better chance of being successful if we keep our best players so that's where I sit on it."
The Cherries then announced the signing of Norwich full-back Max Aarons, hijacking Leeds' move for him, later on Thursday.