Howe guided the Cherries to promotion from League Two last season and is now proving his credentials in the third tier with the club currently in second place, three spots above the managerless Addicks who parted ways with Phil Parkinson last week.
Having supposedly been a target for Southampton earlier in the season, Howe had been heavily linked with the vacant posts at The Valley and Championship club Crystal Palace.
But Bournemouth chairman Eddie Mitchell told Sky Sports News: "We think he has made the right decision and we are 100% behind him. Hopefully we can keep making strides forward.
"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 and rightly so, but hopefully that's the end of it. We are just pleased with the outcome."