The Stadium of Light outfit went into Saturday's fixture having shared the spoils in four of their opening five league fixtures - losing the other - but failed to pick up three points once again in a drab home affair.
Connor Wickham squandered an opportunity to head the hosts in front in the 74th minute, before Angel Rangel was shown the red card for a second bookable offence.
Substitute Steven Fletcher went close to swiftly taking advantage of the extra man - again failing to find the target with a headed effort - but the scoreline remained goalless.
And Poyet bemoaned Sunderland's lack of cutting edge in the final third.
"What do we have to do to win a football game?," he is quoted as saying by BBC Sport. "We did everything. We were the better team 11 versus 11.
"We created chances and we were the team looking to have that winning goal.
"When they had a man sent off it was even more like that but we can't put the ball in the net and take advantage of our game.
"If we were terrible and couldn't pass the ball or create chances I would be worried but the way we played I am calm.
"I will look to prepare to make sure we win the next game."
Sunderland host Stoke City in the Premier League next Saturday.
