Norwich secured a 2-0 victory thanks to first-half goals from Robert Snodgrass and Alexander Tettey.
Sunderland's disappointing day was compounded by the late sending off of Marcos Alonso.
Gus Poyet's side remain in the bottom three and face a tricky test at title-chasing Liverpool on Wednesday.
"We need to be ready to fight and stick together," said Bridcutt. "We have to forget this result and move on and be ready for Liverpool on Wednesday.
"The boss has told us how disappointed he is and we know what we need to do. A positive we can take is that our next game is only a few days away."
The former Brighton man admitted the Black Cats were not expecting such a stern test at Carrow Road and was quick to offer an apology to the club's supporters.
"Of course we are disappointed with the result, it was a big game," he added.
"It turned out to be tougher than we expected and it was one of those days for us - nothing went our way and Norwich were at us from the get-go.
"I'm sorry for the fans who made the trip to Norwich, it is a long, long way from Sunderland and they deserved much more than that.
"Every game from now until the end of the season is massive for us - we need to keep fighting."