The Magpies, who could have gone above the Cherries with a victory on the south coast, produced the best moment of the first half when Venezuela international Salomon Rondon curled in a sublime free-kick in the fifth minute of stoppage-time.
Having seemingly got a dressing down from Eddie Howe at the break, Bournemouth made the perfect start to the second period when referee Mike Dean pointed to the spot in the 48th minute for a foul by Federico Fernandez on Nathan Ake.
Norwegian forward Joshua King calmly slotted home from the penalty to level the scores and the hosts could have gone ahead just two minutes later when centre-back Chris Mepham headed just wide from a corner.
Midfielder Jefferson Lerma then spurned a chance from six yards out before Callum Wilson's looping header from inside the area was superbly cleared off the line by Welshman Paul Dummett.
Bournemouth took the lead in the 81st minute when King fired in a superb effort from the edge of the box after some brilliant hold-up play from substitute Dominic Solanke.
However, with the last kick of the game, Scottish winger Ritchie fired in a sensational half-volley from just inside area to give Newcastle a share of the spoils.