The Canaries knew only a win against Charlton, coupled with defeat for Barnsley at Plymouth, would save them.
As it turned out, the reverse was to happen and a tame 4-2 loss saw Norwich follow the Addicks and Southampton into the third tier for the first time since 1960.
Gunn, who was appointed manager in January until the end of the campaign, said: "The club and the supporters have been let down. To go out on a pitch and capitulate like that is not good enough.
"I want to be involved next season and I will be speaking to the owners and directors over the course of the next few days.
"It's devastating, but we have to fight back and we'll start to fight back tomorrow.
"I need to find out which players want to represent Norwich City in League One next season."
There was a brief moment of hope for City when news of a Plymouth goal filtered through, but the fans' celebrations were cut short by Nicky Bailey's 10th-minute header.
City promptly fell to pieces, and the first two goals of Deon Burton's hat-trick left them 3-0 down after half an hour.
Alan Lee pulled one back before the break, but any hopes of a dramatic escape ended when Burton completed his treble before Sammy Clingan netted City's second with a free-kick.
Gunn led his players to the mass of yellow behind one of the goals after the final whistle, despite chants of 'you're not fit to wear the shirt' ringing out.
"It's a gut-wrenching experience, I had it in 1995 as a player and today was no different," added Gunn.
"I told the players it is one of the worst experiences you can have as a player and they will have to deal with that."
Majority shareholder Delia Smith insisted she had no regrets about giving club legend Gunn the job, and the irony of three teams falling into the third division four years after they were all in the Premier League was not lost on the TV chef.
"He's given his best, he is very enthusiastic and we know him well - so there are no regrets on that front," she said.
"What I do regret is how football has sold itself to money. Now there is one rich league and the rest of us are scrabbling around."
Charlton, whose relegation was sealed a fortnight ago, also have a decision to make regarding their manager Phil Parkinson, who was unable to stop the rot after succeeding Alan Pardew back in November.
"If I'm not here I'll walk away with my head held high," said Parkinson.
"The club has had a lot of instability. If the board, or any new owners, want to start again I think that will be a big risk."