However, after Ritchie's shot had come back off a post, Dwight Gayle was standing in an offside position when he turned the rebound home.
Bournemouth nearly took a surprise lead as Callum Wilson, in his first start for nine months, saw his header tipped over the bar by the acrobatic Rob Elliot before Newcastle's keeper prevented King firing home on the angle from the resultant corner.
Callum Wilson missed a great one on one and Marc Pugh hit the post late on, before Cook's thumping header stole the points for the visitors to lift the Cherries out of the relegation zone ahead of the worldwide break.
It was Atsu's turn nearly instantaneously to have a crack, the former Chelsea winger did well to bring the ball down on his chest before hitting a half-volley from 25 yards but Begović was there again to deny The Geordies an opener.
Newcastle lost captain Jamaal Lascelles, who picked up a knock in the first half, to injury, and at the other end Marc Pugh failed to test Elliot with a presentable header.
The road team won both matches the last time both sides were in the top flight in 2015-16, with Bournemouth winning 3-1 in their last visit to Tyneside.
Wilson - starting a Premier League match for the first time since January - then saw a looping header well saved by a back-tracking Rob Elliot, before the goalkeeper was again called upon to parry King's half-volley.
Howe said: "It was a very hard game for us - Newcastle were good in the first half".
Begovic was kept busy as the game restarted after the interval - the Bosnian palming Joselu's header from a Jonjo Shelvey corner over the bar. The Spaniard's influence proved to be enough as it bounced up off the turf which forced Begović to tip it over the crossbar. King's flick split the home defence and the striker raced onto the pass but fired a left-foot shot into the side-netting when one on one.
It hadn't been the start to the campaign that Howe would have been hoping for as his side entered this clash in the relegation places, but they were rewarded in the final minute of time added on as they stole the only goal of the clash.
However, there was one last moment of drama as Surman's corner from the right was met with a thumping header by Cook - much to the delight of the visiting fans, who had travelled over 350 miles to attend.