Howe was wanted by the Eagles to replace George Burlye, but the Cherries manager confirmed on Tuesday he had rejected the position.
"We think he has made the right decision and we are 100% behind him. Hopefully we can keep making strides forward," Eddie Mitchell, the Bournemouth chairman, told Sky Sports News.
"It's a coup for us to keep him here, we're all very happy with the decision he has made and hopefully we can now draw a line under it.
"It was a big decision for him to make and he didn't take it lightly, but hopefully that's the end of it. We are just pleased with the outcome."
Dougie Freedman is expected to continue as caretaker boss until next week, leaving him in charge for Saturday's Championship trip to Swansea City.
Reading have extended the loan of midfielder Mikele Leigertwood until the end of the season.
Leigertwood, 28, has been on loan from Queens Park Rangers since November.
Darren Ferguson was only sacked as Preston North End manager att he end of last month, but could be set for a quick return at his former club Peterborough United.
He has met with their chairman Anthony MacAnthony. League One Posh are looking for a new manager after parting company with Gary Johnson, little over a year after he replaced Ferguson.
Leeds United have released midfielder Amdy Faye. Faye, 33, made eight appearances for the club after joining on a short-term contract in October.
Leeds have lost an experienced midfielder, but gained a promising young one: signing Zac Thompson, 18, on a free transfer from Everton.