Alexander Isak cancelled out Philip Billing's opener as Eddie Howe's men picked up another Premier League point, with the Magpies having lost just once this season - but now without a win since the opening day.
However, once again they will feel they deserved more as, despite being some way short of their best, they bossed possession and created enough opportunities to warrant victory but could only take one of them.
Bruno Guimarães returned to the starting XI after injury while Dan Burn was preferred to Sven Botman and Loris Karius was on the bench after signing a short-term deal following a training ground injury to Karl Darlow.
Before kick-off, tributes were paid to Her Majesty Queen Elizabeth II, with both managers leading the two sets of players to the centre circle, where Amanda Staveley and Cherries chairman Jeff Mostyn laid wreaths ahead of a minute's silence and a rousing rendition of the national anthem, God Save the King.
But it was a quiet start to the game, with United on the front foot but the first two openings falling to the visitors. First, Ryan Christie had a shot comfortably saved by Nick Pope, then, following a swift counter-attack, former Magpies Academy player Marcus Tavernier sent a header looping just wide.
It took until the 18th minute for Newcastle to fashion a chance of note, with Isak and Miguel Almirón exchanging passes before the latter's tame shot was gathered by Neto, the Bournemouth goalkeeper. An inviting opportunity then fell the way of Fabian Schär, who peeled off his marker to meet Matt Targett's cross after a free-kick was played short, but his header was saved with relative ease.
Midway through the first half, United hit the woodwork for the first time when Guimarães was fouled and Kieran Trippier's free kick beat Neto but struck a post.
Dominic Solanke sent an ambitious bicycle kick wide for the visitors in the 29th minute before, shortly before half time, the Magpies saw three opportunities go begging in an extraordinary passage of play.
Joelinton's shot hit the angle of post and bar before the Brazilian was superbly denied by Neto, then the Cherries goalkeeper dived full length to save from Ryan Fraser seconds later.
Fraser, facing his former club, bent an effort just wide of the far post at the beginning of the second half, then Joelinton headed just over.
But Bournemouth offered a reminder of their threat on the break when Solanke made space with a powerful run, but his shot was straight at Pope.
United continued to try to force the issue, with Joe Willock clipping a ball in from the right but Isak unable to get enough power behind his header to seriously trouble Neto.
However, the South Coast side took the lead in the 62nd minute, when Billing directed Jordan Zemura's left-wing centre beyond Pope.
But Newcastle responded instantly as, less than two minutes later, Trippier's cross was deflected behind by the hand of Jefferson Lerma and, following a VAR check, referee Craig Pawson pointed to the spot.
The Magpies have seen so many decisions go against them in recent weeks, but made the most of their change of luck as Isak stepped up to convert emphatically from the spot, sending Neto the wrong way for his second goal in three games in black and white.
There was a minute's applause in the 70th minute, in honour of The Queen's 70-year reign, before Tavernier fired wide for Bournemouth and Jacob Murphy saw a goalbound effort blocked by Billing.
But despite six minutes of added time, and a late Joelinton header which was saved, it ended all square - a result which suited the away side far more than United, who will hope to have Callum Wilson and Allan Saint-Maximin back after the upcoming international break.
