In a pulsating second half at St. James' Park, the Cherries led through Dominic Solanke after Martin Dúbravka's error but the Magpies hit back seven minutes later through Anthony Gordon, who slotted home a penalty after a long VAR check confirmed a foul on Fabian Schär.
Antoine Semenyo restored Bournemouth's lead with a well-taken drive which looked to have handed the visitors victory, until substitute Ritchie struck from close range in added time against his former club to earn Eddie Howe's men a share of the spoils.
An early flurry of openings for the hosts saw Gordon, the focal point of Newcastle's attack in the absence of Alexander Isak and Callum Wilson, nod a Sean Longstaff cross into the turf and wide before the former Everton man charged down goalkeeper Neto's clearance, with the Cherries stopper fortunate the ball flew wide and didn't ricochet into the net.
But it was Bournemouth who created the clearest chance. After Marcus Tavernier had warmed Dúbravka's gloves with a drive from distance, United's Slovakian custodian produced a fine save to keep out Solanke's low effort after Justin Kluivert had broken free and played him in.
With a quarter of the contest gone, Dúbravka made an even better stop. Lloyd Kelly's centre from the left was inviting and Solanke slid in to meet it in the six-yard box. The Cherries forward looked certain to score but Dúbravka's instinctive intervention was a magnificent one, denying Bournemouth what would, at that stage, have been a deserved deadlock-breaker.
Miguel Almirón whistled one wide from and Schär tried his luck to no avail too before the break, and just after the restart - thanks to a terrific through ball from Longstaff - Almirón had a sight of goal, but lifted his effort over Neto's crossbar.
Just five minutes into the second half, though, Andoni Iraola's men struck. It was a gift from Dúbravka, whose untimely slip in his own six-yard box presented Solanke with an opportunity he couldn't miss.
The visitors were pegged back within a matter of minutes after a protracted VAR check. It centred on a foul on Schär by Adam Smith, who had hold of the Swiss centre back's shirt, and - after an agonisingly lengthy delay while officials studied replays - resulted in referee Michael Salisbury pointing to the spot. Gordon stepped up and confidently converted, lifting the mood inside St. James' Park in the process.
Kluivert and Tavernier both blazed over in the minutes that followed the leveller, while Marco Senesi almost inadvertently put Newcastle in the lead when his sliding block on Bruno Guimarães' cross flashed past the upright, but with just over 20 minutes remaining Semenyo put Bournemouth back in front. It was a superb strike from the ex-Bristol City man, who took Solanke's pass in his stride and unleashed a fine low strike which found the bottom right-hand corner, leaving Dúbravka helpless.
The Magpies keeper had had a solid afternoon but for his slip for Bournemouth's first, and did well to palm Lewis Cook's dipping effort away inside the final ten minutes. It proved important, as in time added on Newcastle hauled themselves level.
Ritchie - who had only been introduced a matter of seconds earlier, bundled the ball home from close range with his first touch of the ball after his own header had rebounded back into his path via a Cherries defender and Neto, snatching a point for the hosts after a thoroughly breathless second half on Tyneside.
