When City took the lead through Fernandinho in the second minute, Sunderland were at risk of being blown out of the tie before it had even started.
However, a second-half brace from Connor Wickham saw Sunderland snatch the initiative away from home.
City hurried in response and shared the spoils thanks a late Samir Nasri goal, which Vito Mannone appeared to stop before pushing into the corner of his own net.
A point leaves Sunderland six points adrift of Premier League safety, with a game in hand on their closest rivals, but their Uruguayan coach is not willing to throw in the towel.
"Survival was the challenge from the beginning and it is still possible," Poyet told Sky Sports.
"We will fight, there are too many rumours and too much stupidity. Too many people talking but at the end of the day I will be here."
Poyet was dismayed to see his side written off before the game and was quick to acknowledge his critics after holding the often free-scoring City to a stalemate.
"There will be some genius who says, now we drew, that they were expecting us to fight and do something incredible, but he will lie," he claimed.
"Nobody did anything for us today, everybody wrote that we were getting demolished for whatever reason .
"We played and if there is one thing top teams do not like it is to defend."
After City's earlier opener, Sunderland enjoyed a period of dominance and John O'Shea was twice in space to head but squandered both chances, while Fabio Borini was also sent through on goal.
"I think the first half we were even better than the second half, the only problem is we didn't score," Poyet said.
"At half-time, you need to convince the team at the bottom that we are the best team and to keep believing.
"The players tried their best and we were concentrating on making it difficult for them and trying to go forward and create chances."
Sunderland levelled the scores with just over 15 minutes to play when Emanuele Giaccherini's cross beat Joe Hart, leaving Wickham to tap home from close range.
The Sunderland striker doubled his tally 10 minutes later after a fine through-ball from Borini and Poyet praised the 21-year-old, who was on loan in the Championship earlier this season.
"We had a player today in Connor that gave us something we were missing for a long, long time," he added.
"A couple of goals in one game is something that will give you points.
"I don't want to take about individual players or actions, I think the most important thing is the belief of the players; they were calm on the ball and they wanted to show everyone we are a better team than the table shows."