Despite having more possession and the lion's share of the chances, the Black Cats couldn't break the deadlock and missed an opportunity to secure a vital three points.
And Poyet was left to rue what might have been as his side remain rooted to the foot of the Premier League table.
"We deserved to win the game. We don't get any breaks, we are getting nothing," he told Sky Sports.
"I said it two weeks ago when we played Villa, and this was away.
"Today from the first minute we were having chances, having shots, one-on-ones, and when you are at the bottom, you need to take those chances.
"Today we have 19 shots, and how we don't score, I don't know."
The result means that Sunderland remain bottom of the league, and are without a win in their last five matches.
"We need the wins playing like that. I am waiting for an ugly win," Poyet added.
"When we play well, and don't win it is a little bit tough, but I need to be pleased with the way we played."