The Magpies were trailing to Harry Wilson's well-taken opener at St James' Park, but DeAndre Yedlin levelled with a powerful header with the break approaching and Republic of Ireland defender Clark, who scored in last Saturday's 3-2 victory at West Ham, struck seven minutes after the restart to seal a 2-1 win.
However, although they continued to threaten on the break, the hosts also had to defend tenaciously in front of a crowd of 44,424 at St James' Park as the visitors fought all the way to the whistle in an ultimately vain mission to get something from an entertaining game.
Steve Bruce's men head into the international break with 15 points from their first 12 games, a more than satisfactory return from a difficult run of fixtures, and sit just a point behind the Cherries, who started the day in seventh place in the table.
Newcastle started brightly, but it was the visitors who mustered the first attempt on goal when Ryan Fraser blasted a dipping sixth-minute effort just over Martin Dubravka's crossbar following a slick interchange of passes between Josh King and Adam Smith.
Dubravka had to be alert to claim King's driven ninth-minute cross after Smith and Callum Wilson had carved the Magpies open down the right, but it was Bournemouth goalkeeper Aaron Ramsdale who had to turn away Jetro Willems' firm drive seconds later after he had got in behind Smith.
However, Eddie Howe's side took the lead with 14 minutes gone when a well-worked short corner move saw Fraser receive the ball back from King and square for Harry Wilson to fire home.
It might have been two had Dubravka not managed to repel full-back Diego Rico's ambitious effort at his near post three minutes later and, although Jonjo Shelvey and Joelinton both took aim from distance without troubling Ramsdale, it was the visitors who remained largely on top.
Newcastle needed Yedlin to clear Fraser's attempt off the line after he had beaten Dubravka to a ball over the top, but they should have been level seven minutes before the break when Allan Saint-Maximin carved his way into the box but fired against the bar from point-blank range, and Ramsdale had to save from Willems in the aftermath.
But there was nothing the keeper could do to keep out Yedlin's bullet header from Saint-Maximin's 42nd-minute cross in a breathless finish to the half.
Newcastle went ahead within seven minutes of the restart when Shelvey, Saint-Maximin and Willems worked a planned move from a corner and Fernandez headed on for Clark to stab home.
Ramsdale had to get down well to save another Willems strike after a thrilling 58th-minute run by Saint-Maximin and then claw away Joelinton's lofted attempt, and Miguel Almiron was denied his first strike for the club 12 minutes from time when his goal-bound shot hit the prone Saint-Maximin with the net yawning.
King might have snatched a draw six minutes into stoppage time, but headed wide from beyond the far post after Fraser's corner had been flicked on.