The Bosnian striker set the visitors on their way to a 10th successive win over the Magpies with an early finish and Negredo struck in added time as City extended their unbeaten run in all competitions to 15 games.
However, the home side were hugely aggrieved to see a 34th-minute Cheick Tiote equaliser ruled out by referee Mike Jones for offside.
The Ivory Coast international blasted home from distance, but after speaking to assistant Stephen Child, Jones ruled out the goal - to the fury of Newcastle boss Alan Pardew, who later became involved in an angry touchline spat with opposite number Manuel Pellegrini.
Newcastle were seeking to avoid a fourth successive defeat but the early signs were not good as City, fresh from their 6-0 Capital One Cup semi-final first leg demolition of West Ham in midweek, ripped into them from the off with full-back Aleksandar Kolarov terrorising counterpart number 13 Mapou Yanga-Mbiwa.
It took them just eight minutes to go ahead when David Silva played Kolarov in behind Yanga-Mbiwa and his driven cross was thumped home by Dzeko.
It could have been worse for the Magpies with the in-form Negredo just failing to connect with another Kolarov cross in front of goal and then looping a 15th-minute header on to the roof of Tim Krul's net.
But with Yohan Cabaye, who had started in a number 10 role behind lone striker Loic Remy, dropping deeper, the home side took a foothold in the game and gradually stemmed the tide.
Defender Steven Taylor stabbed a Cabaye corner wide, and the French international forced a fine 33rd-minute save from England keeper Joe Hart with a rasping shot from a near-impossible angle.
But the resulting corner brought controversy as Newcastle thought they had dragged themselves back into the game.
The ball was cleared to Tiote on the edge of the box and he unleashed the sweetest of left-foot strikes which left Hart rooted to the spot as it screamed into the back of the net.
The midfielder, whose last goal for the club was a spectacular long-range equaliser in a 4-4 draw with Arsenal in February 2011, set off on a joyous run, unaware that Jones was consulting his assistant.
Replays confirmed that Yoan Gouffran was one of three players in an offside position, and the pair decided the Frenchman was interfering with play, a view not shared by the vast majority of the 49,423 spectators inside St James' Park.
Pardew was certainly of that persuasion and he and Pellegrini were involved in a heated exchange as the whistle approached.
Buoyed by a sense of injustice, the Magpies returned in determined mood and Vurnon Anita forced a 48th-minute save from Hart before Remy was presented with a glorious opportunity to level.
Moussa Sissoko's pass put the striker in on goal, but as he tried to turn inside Vincent Kompany rather than shooting first time, he took a touch too many and the chance had gone.
However, City were far from done and Krul had to make a fine 55th-minute block to keep out a point-blank header from Alvaro Negredo, who had earlier fired high over on the turn.
But Hart had to dive to his right to keep out Cabaye's swerving 59th-minute attempt and then punch away the same player's free-kick two minutes later with the home side dominating.
The former Lille midfielder was waging a one-man war on Hart's goal and sent another dipping shot just over with 68 minutes gone, but it was Remy who was denied by the goalkeeper after latching on to Gouffran's cross two minutes later as Kompany and Pablo Zabaleta collided.
As Newcastle pushed forward, City exploited the space they left behind and Fernandinho rattled Krul's crossbar with an 87th-minute header.
Remy might have snatched a point with a header which sailed just high and wide from Cabaye's corner and Taylor saw an injury-time shot blocked by Dzeko.
However, Negredo made sure City would displace Chelsea - 2-0 victors at Hull yesterday - deep into the 95th-minute, despite Krul blocking his initial effort.