The Swansea winger landed awkwardly on his right ankle following a hefty challenge by City captain Sebastien Bassong in the 1-1 draw.
An early prognosis suggested the 26-year-old had sustained a fracture and Dyer said: "I feared the worst.
"My ankle twisted, gave way and I felt a big crunch. I was convinced it was broken. I was in tremendous pain and had to have a lot of morphine waiting for the ambulance to come.
"I was feeling good in the game, especially after getting the early goal."
Hospital x-rays confirmed no break but Dyer still faces a number of weeks in rehab with ligament damage on his right ankle.
"I suppose it's one of mixed emotions. I'm relieved the injury is not as bad as I first thought, but I still hate being out and doing rehab," he added.
"I always like to keep positive though and there's been worst injuries than mine.
"I've had some nice messages from Swans fans and those at other clubs. Bassong and Norwich manager Chris Hughton both sent me a text after the game to wish me well, while Bradley Johnson, who's a good friend, also phoned me in the hospital.
"Their concern and kind words were appreciated and now I just want to get back playing as quick as I can.
"I've scored five goals so far this season and I've got a target to reach - although I'm not telling anyone what it is!"