The former Brighton boss was left unimpressed by his side's performance in a tightly contested affair at Turf Moor, with chances at a premium at both ends.
Patrick van Aanholt rattled the post with a late long-range drive, but that was about as close as the visitors came on a frustrating day for both sides.
And Poyet was the first to admit that his side had not done enough to come away from Lancashire with all three points.
"We played okay, but okay is not enough in the Premier League - you need to play better than okay. We were defending okay, passing the ball okay and we need to be better than that," he said.
"I was expecting a little bit more. I wanted to control the game more, I wanted us to be in better positions and create better chances so it was one of those days, where everything we planned didn't go as well as we would have liked."
The Uruguayan then warned his side that any similar lacklustre performances could prove costly in future games, while also implying that the trip to Turf Moor represented a chance missed with Sunderland still searching for their first Premier League win of the season.
"You cannot play 'okay' because you can get punished. Even one action or one deflected shot can go in and you lose the game," Poyet added.
"We needed to impose certain things today and with all due respect to Burnley, these are players playing at the highest level, they cost a lot of money and these are the games where they need to stand up.
"I think it was a great opportunity to take three points today and we missed it."