It was some poor goalkeeping from Begovic that gave Newcastle a 2-0 lead, before Bournemouth rallied to get a point in the game.
But the keeper's manager is confident that he will be back to his best soon enough.
Howe said: "He is an outstanding goalkeeper and has had a really good season for us.
"The life of a goalkeeper is a difficult one and any mistake you make is highlighted.
"But he has been good for us this season so I back him to recover.
"With a player of Asmir's experience, I don't think there is a lot you can do or say, it comes from within.
"He has had many experiences in his career, many highs and lows.
"It's not the first mistake and it won't be the last he makes in his career. That's what a goalkeeper has to endure.
"But what he has always done for me, he has always come back from difficult moments and produced outstanding performances.
"That's the mark of a true character, a leader within your group. Asmir inspires confidence in his back four so we are very happy with him."