Cunha put Nottingham Forest ahead in the sixth minute before Delap doubled their advantage on 18 minutes. Rayan Cherki pulled one back shortly after half-time, but Delap struck again in the 73rd minute to secure a memorable victory.
Oliver Glasner was delighted with the response from his players, who had been challenged to raise their standards.
"You can never plan to win 3-1 against City because they are so good. But today, especially the players and our fans deserve this win because they were so passionate from the first minute, scoring nice goals.
"We had moments to suffer but then we could rely on our fans pushing us, supporting us. Amazing goals and such a great team effort today so really proud.
"I think we've had better performances in possession because today we rarely had the ball. Regarding the team spirit, the energy, I think it was our best game. The way we defended, how passionately we defended, how tactically smart we defended, it was one of our better games."
Glasner believes matching Manchester City's intensity was essential.
"I think it is the only way you can beat Man City. They're so good, such great players, such a good structure and they move the ball so quickly. They can play through the middle, they have incredible dribblers on the wing and they have such pace. Then, of course, they have Haaland in the middle.
"You should control everything and if you want to control it, you have to run."
The Forest manager was also pleased with Delap's contribution, saying: "These goals are his biggest strengths with his right and his left, also using his power and his strength. We knew that he would score goals for us and today, they were nice ones."
Manchester City boss Enzo Maresca admitted his side had failed to match their opponents.
"First half when you don't match intensity and lose all the duels it's difficult. Then when we want to react, it's too late.
"In this kind of match, you need to match intensity and win duels, otherwise it's always complicated. First half we were not good enough in terms of duels, second balls - we lost all of them."
Maresca explained that Marc Guehi requested a half-time substitution because of an injury, while Josko Gvardiol had also been carrying a problem during the week.
"Second half was better but we don't need to wait to concede one or two and then react. We need to start from the beginning."
City have now lost three consecutive matches after winning their opening seven, and Maresca acknowledged that his team still have work to do.
"It's a learning journey. We know what we do well but we need to understand what we are not doing well and try to improve it."