The Magpies fell behind before the half-hour mark at St. James' Park following Bernardo Silva's back-heeled finish before two strikes within the space of two minutes, courtesy of Alexander Isak and Anthony Gordon sent the hosts ahead before the half-time break.
Kevin De Bruyne's stroked effort levelled the contest, five minutes after the Belgian's introduction from the substitutes' bench, before Newcastle were delivered a sickening late blow as Bobb placed the ball home following De Bruyne's perfectly weighed pass in the final minutes of a pulsating English top-flight encounter.
Eddie Howe made one change from the side which claimed a 3-0 victory against Sunderland in the Emirates FA Cup last weekend as Lewis Miley, who made his full competitive debut for his boyhood club in September's Carabao Cup win against City, replaced the injured Joelinton in the middle of the park.
Inside the opening two minutes, Sean Longstaff looked to have fired the hosts in front after slotting the ball home from Alexander Isak's pass but the latter had strayed into an offside position in the build-up.
Newcastle's positive opening continued, with Miguel Almirón blocking Ederson's intended clearance before Anthony Gordon's pass found its way to Bruno Guimarães but the Brazilian midfielder scuffed his attempt at goal before the ball was cleared by the City defence.
City, who won the FIFA Club World Cup last month, went close to breaking the deadlock as Silva capitalised on a loose ball inside the penalty area but directed his half volley over the crossbar.
Silva, however, opened the scoring in the 26th minute after the Portuguese forward cleverly flicked the ball into the bottom left-hand corner from Kyle Walker's low cross, with the home crowd left frustrated beforehand after believing Gordon was fouled by Rúben Dias prior to the goal.
Magpies goalkeeper Martin Dúbravka produced a vital save to prevent City doubling their lead in quick succession, with the Slovakian shot-stopper crucially tipping Silva's powerful volley onto the woodwork.
Isak went close to netting his 10th Premier League goal of the 2023/24 campaign minutes later as the 24-year-old striker blocked Stefan Ortega's intended pass before latching onto Almirón's resulting header but the Swede's shot at goal was blocked by Rodri.
In the 35th minute, Newcastle levelled the scoring through Isak, netting his fifth goal in his last four competitive matches for the Magpies. After strong defensive work from Fabian Schär, the Swiss defender played the ball into Guimarães' path before the South American delivered an excellent first-time lofted pass into Isak's path, who fired a terrific effort into the right-hand corner.
Two minutes later, Gordon sent United into the lead with his seventh league goal of the season as the Liverpool-born forward raced onto Burn's pass before the 22-year-old winger drove into the box and drilled a fierce effort into the right-hand corner.
Newcastle went close to doubling their lead before the half-time whistle through Isak, who raced towards goal after Miley's through pass but his resulting strike was blocked by Ortega.
In the second-half, Dúbravka was forced into early action as the experienced goalkeeper, celebrating his 35th birthday on Monday, beat away Álvarez's long-range free-kick before holding Foden's low drive from the edge of the penalty area shortly after the Argentine's set piece.
The United 'keeper recorded another important save to prevent City from equalising as Mateo Kovačic's deflected drive forced the Slovakian to readjust his feet before holding the Croatian's effort from the edge of the box.
Álvarez, who netted the winning goal in the reverse fixture against the Magpies in August, almost brought Pep Guardiola's side on level terms as the ball landed kindly into the striker's path inside the box but the 23-year-old drove his attempt over the bar.
In the 74th minute, Newcastle found themselves on level terms as De Bruyne collected Rodri's pass before the two-time Premier League Player of the Season slotted a low effort into the bottom right-hand corner.
Dúbravka prevented Rodri from sending City ahead in the latter stages as the Spaniard dangerously advanced into the penalty area before the midfielder's strike was pushed out for a corner by the goalkeeper's left hand.
With three minutes of stoppage-time awarded, however, Newcastle suffered an agonising blow as Bobb sent the visitors ahead, latching onto De Bruyne's pass in behind the United defence before the 20-year-old Norwegian produced quick feet to evade the incoming Dúbravka before tucking the ball into an empty net.
