Both sides hit the woodwork in the closing moments, Clarets substitute Ashley Barnes sending a deflected effort against the crossbar after 87 minutes and Black Cats defender Patrick van Aanholt rattling the post in added time.
But that late drama was at odds with an otherwise dour encounter at Turf Moor that seemed locked in first gear for long periods.
On this evidence it is not hard to see why both teams are still awaiting a first win of the campaign.
It may be tempting for the hosts to view things positively, a third goalless draw in a row hinting at a useful defensive resolve, but Poyet was unimpressed.
"The clean sheet I can take from today, nothing else," he said.
"I was expecting a little bit more
With all respect to Burnley, our players play at the highest level, they cost a lot of money
These are the games they need to stand up in - as a team and as individuals.
"I said to them before coming here, 'we're playing okay'.
"But okay is not good enough in the Barclays Premier League
You need to be better than okay.
"Here we defended okay, we passed the ball okay but we need to be better than that."
Asked what a season of such football would bring, Poyet added: "One point a game, 38 points maybe.
"You cannot just be okay
You are going to get punished.
"I think if we had been a little bit better this was a great opportunity for three points and we missed it."
Aside from Van Aanholt's last-gasp attempt, Sunderland had little joy going forward.
Connor Wickham looked more like the striker loaned to Sheffield Wednesday in the early part of last season than the one whose goals inspired a great escape in the closing games, while Adam Johnson was worryingly peripheral.
It was a similar story of frustration for Burnley, who have not scored since their opening-weekend defeat by Chelsea.
The absence of Danny Ings did not help against Sunderland, with stand-in Marvin Sordell wasting a fine early chance to break the deadlock.
But manager Sean Dyche believes things are heading in the right direction.
"We're very pleased with the clean sheet again - that's hard to do in itself, three of them back to back
But I'm very pleased with a lot of our attacking play too," he said.
"We found a better balance today
We looked solid but we were creating chances.
"If we keep playing like that I'm absolutely convinced there are wins around the corner for us.
"I know you have to create chances and take them but if you create enough they'll start going in eventually.
"I'm not naive - I know you've got to score goals to win games but you've also got to believe in performance levels and ours are good at the moment."
Source : PA
Source: PA