Manuel Pellegrini's men, who were forced to substitute both Stevan Jovetic and Micah Richards due to injury, rallied in extra-time to book their place in the quarterfinals
Newcastle will feel hard done by after taking the match to Manuel Pellegrini's side, but it failed to find a breakthrough and was left to rue goals from Negredo and Dzeko that put City into the quarterfinals.
Alan Pardew's men created the best chances of the first half and the home team thought it had the lead through a close-range Shola Ameobi effort, but he was adjudged to be offside.
City – which last won the competition in 1976 - improved after halftime with former Newcastle man James Milner coming closest with a long-range curler. Papiss Cisse wasted a great chance to put Newcastle ahead in extra time when he volleyed straight at Costel Pantilimon.
And Newcastle was duly punished by Negredo's side-footed effort from Dzeko's low center, before the Bosnia-Herzegovina international added gloss to the scoreline when he finished Milner's sublime pass.
Both teams took the opportunity to rotate their squads, Ameobi made a rare start for Newcastle, while Pantilimon replaced under-fire goalkeeper Joe Hart for City.
In a lackluster opening, Newcastle came closest to opening the scoring when Ameobi flicked on Tim Krul's long pass and Cisse lofted his effort over the goalkeeper and just wide of the post.
Some stout defending gave Newcastle a solid base and it was nearly rewarded on the half-hour mark when Ameobi's half-volley forced Pantilimon into a smart one-handed save.
Pardew's side thought it had taken the lead with the next attack when Ameobi tapped home Cisse's center but his celebrations were curtailed by the offside flag.
After struggling to make an impact in the first half, City came out with more intent after the break and nearly went ahead when Jesus Navas dispossessed Krul and laid back to Milner, who curled his 25-yard effort at the open goal just wide.
City was now in the ascendancy and Negredo wasted a golden opportunity to grab the winner. The Spain international received Dzeko's lay-off in the area, but took too many touches before chipping tamely at Krul.
That would prove the best opportunity of the closing stages as the match headed for extra-time.
Newcastle made a fast-paced start after the restart and Cisse missed his side's clearest chance of the game when he sprung the offside trap only to shoot straight at Pantilimon.
That miss would prove costly as City edged ahead in the ninth minute of extra-time.
Dzeko broke down the left and crossed low for Negredo to calmly slot past Krul.
And, in the 105th minute, the game was put beyond doubt. Milner drove at the Newcastle defense and played a superb pass with the outside of his foot to Dzeko, who rounded Krul and slotted into the empty net.
Newcastle almost grabbed a consolation when Joleon Lescott cleared Hatem Ben Arfa's driven effort off the line as City booked its place in the last eight.