The Clarets said that the fee for Gibson equals the club record transfer paid last year for striker Chris Wood and that Gibson had signed a four-year contract.
And Boro boss Tony Pulis confirmed after the 2-2 draw at The Den: "He's going, it's agreed, it's done and we want to thank Ben, an absolutely fantastic professional".
1:20 Pulis refuses to rule out Gibson exit
Burnley had failed with two previous bids of £12m and £13m over the past 48 hours but an agreement over a fee for Gibson has now been reached. If he has an opportunity to go to the Premier League then brilliant for Ben.
Gibson would be a big miss for Middlesbrough having impressed at the Riverside Stadium both in the Championship and the top-flight, but a move for him was inevitable after they missed out on promotion last term.
"I've played for Middlesbrough for 15 years since I was a boy so it's a huge step" he added.
"I'm sure the club he's come from want to challenge to get in there, but we are now there, and it's a new start for him, someone who's been at the club for a while". After Burnley made Gibson their number one target, it was always likely the Boro vice-captain was set to leave his hometown club for Turf Moor.