The 25-year-old -- capped at all age levels by England up to Under-21 -- signed a four year contract after he stayed loyal to Middlesbrough when they were relegated after just one season in the Premier League a year ago.
However, Gibson -- nephew of Middlesbrough chairman Steve Gibson -- has been persuaded to move with the attraction of a potential Europa League group campaign if Burnley come through the qualifying rounds and 'Boro having failed to regain their Premier League status.
"Burnley Football Club is delighted to announce the signing of defender Ben Gibson from Middlesbrough," read a statement from the club on their website.
"Gibson has agreed a four-year-deal at Turf Moor after Burnley paid a joint club record fee for the centre-back."
Gibson's transfer fee matches that paid by Burnley to another second tier outfit Leeds United for New Zealand international striker Chris Wood last year.
He is the club's first signing of the close season, the transfer window closes next Thursday.
Former England first choice goalkeeper Joe Hart had been due to undergo a medical on Friday with a view to moving from Manchester City, where he has been out of favour since Pep Guardiola took over two years ago and has been sent on loan firstly to Italian side Torino and then last term to West Ham.
Burnley face Turkish side Istanbul Basaksehir in a Europa League qualifier on Thursday, before their Premier League opener against Southampton on Sunday.