The Riverside club were left disappointed by the decision of the Leeds-born player and his representatives to push for a move last week.
Tavernier was offered a new deal earlier in the summer by the Teessiders.
Speaking ahead of Boro's opener with West Brom, boss Chris Wilder said: “We have to get on with it, the club did everything to try and keep Marcus.
"When you get down to the nitty-gritty of the move and him wanting to move, I think it just became a case of negotiating the transfer fee, which is above me.”
"But as I say the club tried everything they could to keep Tav, and now we have to look forward."
On completing his move, Tavernier, 23, who has signed a five-year deal with the Dorset club - with Boro receiving some add-ons alongside the initial £12m fee - commented: "I'm happy to be here, it's been a long time coming. I'm sure everyone has seen in the press it's been something that's been going on in the background, and I'm just delighted to be here now."