The Magpies looked to be heading for a home defeat when Antoine Semenyo sent the Cherries 2-1 ahead in the 69th minute of the contest, but Eddie Howe's side would not be beaten.
The result has left Newcastle seventh in the table, four points behind sixth-placed Manchester United, while Bournemouth occupy 13th, nine points clear of the relegation zone.
Newcastle's first half-chance of the match came for Anthony Gordon in the fifth minute, but his header at the back post was always dropping wide of the target.
Neto had a scare in the Bournemouth goal three minutes later when he took too long to make a clearance, with Gordon charging it down, but the ball bounced wide of the goal, much to the goalkeeper's relief.
Bournemouth should have scored in the 16th minute when Justin Kluivert worked his way into the Newcastle box before finding Dominic Solanke, but the striker was unable to beat Magpies goalkeeper Martin Dubravka.
The Cherries had another big chance in the 24th minute, again through Solanke, with Marcus Tavernier finding his teammate on this occasion, but Dubravka made another smart save.
Neto then had to keep out a strike from Gordon down the other end during an open period of the match, with Miguel Almiron also missing the Bournemouth goal with a rasping effort.
The breakthrough arrived in the 51st minute, with Solanke finding the back of the net after a huge mistake from Dubravka, who had taken too long on the ball and was caught out.
Newcastle were awarded a penalty soon after following a lengthy VAR check, with Fabian Schar adjudged to have been fouled inside the box, and Gordon stepped up to level the scores at St James' Park.
Bournemouth managed to regain their lead in the 69th minute, though, with Semenyo finding back of the net with an excellent strike from the edge of the area after being released by Solanke.
Dubravka was forced to push away a strike from Lewis Cook in the period that followed, with Bournemouth looking for a third, but Newcastle managed to incredibly level the scores in stoppage time through Ritchie.
There was chaos inside the Bournemouth box when Bruno Guimaraes delivered a dangerous cross, and Ritchie was on hand to poke into the back of the net to spare the home side's blushes at St James' Park.