Newcastle will feel hard done by after taking the match to Manuel Pellegrini's side but they failed to find a breakthrough.
Alan Pardew's men created the best chances of the first half and thought that they had the lead through a close-range Shola Ameobi effort but he was adjudged to be offside.
City – who last won the competition in 1976 - improved after half-time with former Newcastle man James Milner coming closest with a long-range curler. Papiss Cisse wasted a great chance to put the hosts ahead in extra time when he volleyed straight at Costel Pantilimon.
|VIEW FROM ST JAMES'S PARK
|By Michael Lightfoot
Newcastle can count themselves unlucky as they were the best side in the first half and for the majority of the second. It was not until extra time that they were finally undone, first by Alvaro Negredo and then Edin Dzeko.
The real story, though, was Costel Pantilimon. Joe Hart must be seriously worried for his position after a superb display from the Romanian, assured and untroubled throughout.
Both teams took the opportunity to rotate their squads, Ameobi making a rare start for Newcastle, while Pantilimon replaced under-fire goalkeeper Joe Hart for City.
In a lacklustre 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 rgw Magpies a solid base and they were nearly rewarded on the half-hour mark when Ameobi's half-volley forced Pantilimon into a smart one-handed save.
Pardew's side thought that they had taken the lead with the next attack when Ameobi tapped home Cisse's centre 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 were 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 breaking down the left and crossing low for Negredo to calmly slot past Krul.
And, in the 105th minute, the game was put beyond doubt. Milner drove at the Newcastle defence 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 but Joleon Lescott cleared Hatem Ben Arfa's driven effort off the line as City booked their place in the last eight.