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Jair Cunha and Delap had given Forest a two-goal lead at half-time - both courtesy of rudimentary City errors - before Rayan Cherki pulled one back with a stunning half-volley.
City appeared to have the momentum but Forest remained dangerous on the counter-attack and Delap produced a spectacular finish to stymie any hopes of a comeback.
The result left City six points behind table-topping Arsenal - 1-0 winners over Hull earlier in the day - in second place, while Forest are now two points above the relegation zone in 16th. It also pushed Manchester United into the relegation zone for the first time at this stage of a Premier League season.
Forest had taken an early lead when Josko Gvardiol was punished for a slightly heavy touch before being bundled off the ball far too easily by Cunha.
The Brazilian advanced towards the edge of the box before stroking a low effort into the corner of the net from just inside the area.
The intensity of Oliver Glasner's side brought them a second goal when Xaver Schlager's tenacity allowed him to win a 50/50 challenge against a timid Elliot Anderson.
Schlager released Delap, who powered his way from the halfway line - unchallenged - before lashing a powerful finish beyond a hapless Gianluigi Donnarumma.
City had their moments during a frenetic first period - Erling Haaland's shot from a tight angle was cleared off the line while Enzo Fernandez should have done better with a header.
The visitors reduced the arrears early in the second half when Cherki gracefully met a clearance with a crisp half-volley.
City then camped in Forest's half but Delap applied the coup de grace when he finished off a swift breakaway with a rasping, curling finish high into the corner of the net.
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