Aaron Wilbraham headed the Robins into an early lead before a defensive error allowed David Cotterill to make it 2-0, as the hosts endured a torrid first half.
But a close-range finish from Jonjo Shelvey got Newcastle back into it after the break, before Ciaran Clark levelled matters and earned a point for the hosts after 81 minutes of what was the proverbial game of two halves at St. James' Park.
After a cagey opening few minutes, Aleksandar Mitrovic glanced a Shelvey free kick into the arms of Fabian Giefer in the first on-target attempt of the afternoon.
But just 11 minutes in, the Robins took the lead. After Bobby Reid had seen his shot charged down, Lee Johnson's side kept their foot on the pedal, with Gary O'Neil sending an excellent, curling cross in from the right.
It was met by veteran frontman Wilbraham, who planted his header across Karl Darlow and into the far corner to put the visitors in front.
There was a further warning for United on 18 minutes as Wilbraham's flick put Reid in the clear, but he delayed his shot, allowing DeAndre Yedlin to get back and clear the danger.
And just two minutes later, the Robins made it two. Paul Dummett's header towards goalkeeper Darlow wrong-footed the former Nottingham Forest stopper, and Cotterill pounced to tap into an unguarded net and give his side an unlikely two-goal lead.
Shelvey whipped a low free kick into the area and just past the post before Matt Ritchie saw his drive deflected over as the Magpies responded with urgency. Then, some excellent play from Yedlin and Ritchie down the right culminated in the latter delivering a low cross, but again, Bristol City survived.
On-loan winger Christian Atsu broke into the box on the half hour but elected to square it, and Perez saw his goalbound effort deflected past the post as United ended the half with a mountain to climb.
After the break, Benitez's men rallied, with Perez lashing just over the bar after Jamaal Lascelles had found him with a pass forward.
But Bristol City remained a threat, and Cotterill's low, swerving effort required a smart stop from Darlow, who parried and then scrambled the ball away at the second attempt.
Just before the hour, though, United made a breakthrough. Perez and Atsu combined down the left, with the Ghanaian winger crossing low into the middle where Shelvey was able to force the ball over the line from close range to get his side back into the contest.
A stunning last-gasp sliding challenge from Aden Flint denied Perez after Mitrovic's clever flick-on sent him clear, with Newcastle now firmly in the ascendency. Ritchie stung the palm of Giefer with a vicious drive, and Mitrovic's header from a Shelvey corner skimmed the bar as United piled on the pressure.
And, deservedly, that pressure eventually told. After Atsu had lashed a shot just past the post, a Ritchie corner found Ciaran Clark, who - perhaps with a little help from a Robins defender - found the target to draw United level.
Shelvey almost won it shortly after with a powerful strike from the edge of the box, while Mitrovic saw another attempt charged down and cleared in the closing minutes as City held on to claim a point despite a much-improved second half display from Benitez's men in front of a crowd of over 52,000 at St. James'.
