"It is just a fantastic opportunity for Jason and myself to join such a big, ambitious and progressive club and we are delighted," added Howe.
"We had such a great time at Bournemouth and it was always going to take a special club for us to leave because of the fantastic success we have had there in the past two years.
"We just felt Burnley was that club and we are ready to take on a huge challenge.
"However it is one we are excited by. The team is in a decent league position but there is still a long way to go."
Howe took charge of the Cherries for the last time on Friday as his side went down 2-1 at Colchester.
Howe, who replaced Jimmy Quinn when he took charge of Bournemnouth on 31 December 2008, said on Tuesday that he intended to stay at Dean Court after being linked with vacancies at Charlton and Crystal Palace.
But he informed his players on Friday morning that he was planning to join the Clarets after their game at Colchester - his 100th in charge of the club.
He told BBC Radio Solent: "I turned down two other clubs this week because I wanted to stay but this was just one I couldn't turn down.
"There were no reasons why I should from a personal and professional point of view. It just ticked all the boxes.
"I found it very difficult from the week's events to come to that decision but I thought if I didn't it would be something I would regret for the rest of my life."
Howe, who has previously rejected approaches from Peterborough and Southampton, was the youngest manager in the Football League when he was appointed by the Cherries.
And, despite being hampered by points deductions and transfer embargos, he took the club from the foot of League Two to their current position of fourth in League One.
Of his time at Bournemouth, Howe said: "We've had two years of unbelievable success here and I hope the lads continue that and go on to get what they deserve. I leave with great sadness."
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