The 23-year-old centre back had a transfer request turned down by Swindon but eventually a bid was accepted from the newly-relegated side before terms were agreed and the defender passed a medical.
Flint made 79 appearances since moving to The County Ground from Alfreton in January 2011 and his final game for the club saw him head home a dramatic late equaliser in the play-off semi-final second leg at Brentford.
Flint has signed a four-year contract and Robins head coach Sean O'Driscoll said: "I saw him play for Alfreton a while back, but in the last 18 months he's improved immensely. I've seen him four or five times since then and he's grown on me.
"He's worked under Paolo Di Canio, through a successful period, and then with Kevin MacDonald, who got Swindon into the play-offs.
"He's not the finished article and we're not bringing in an established centre-half, but he's that type who we can bring in and develop.
"I wouldn't want to class him as a 'head it and kick it' type, because he's not afraid to have the ball at his feet and use it sensibly.
"Any lad who's that big can look a bit cumbersome at times, but Aden and Darren Ward were a good partnership at Swindon.
"He's quick enough too and in terms of looking for a young centre-half who you can actually play straight away and you feel there's improvement to come, he fits the bill."