The Trotters had a £250,000 bid rejected recently and returned with the increased offer at the weekend, triggered by a fear that he could become cup-tied.
"For Bolton it was important to end the deal before the Champions League qualifier against Astana tomorrow as they want to use Braaten in the UEFA Cup later on," read a statement on the Norwegian club's website.
Rosenborg sporting director Rune Bratseth added: "This is of course a loss, but we have no choice. He would have gone away for free after this season, therefore we can't say no."
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Norway international Braaten, 25, was a target for Hibernian, Fulham, Everton and Aston Villa as well as two Ligue 1 clubs. His contract was to expire in January.
"This is football: players come and go," the player told the club's website. "But I will really miss all the kindness in the RBK family."
Braaten admitted that he is a little ignorant about the Lancashire town he will now inhabit.
"I don't know much about the place, but I do know some of the players and a little bit about the club," he told newspaper Dagbladet.
"It seems exciting: definitely a step up."
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