Both teams were looking for their first wins of the season but both had to settle for one point and visiting boss Poyet said: "I don't think it was the best game that you will see in the Barclays Premier League.
"It was tight, there were periods in the first half that I thought we started to play the way I want, using the space for our advantage but it was only glimpses and for a few moments.
"We came back a little bit later so it was in and out. You need to be on it all the time, trying to do the right things. I would say it was an okay performance, but no more than that.
"I think that the final ten minutes showed that we wanted to win the game, both teams needed to win and it was close, but the longer the game went on it was a typical 0-0 or 1-0 maximum.
"I don't think there would be more than one goal and we were trying to make sure it as us who got it. It wasn't to be but we got our first clean sheet in the Barclays Premier League.
"We need to keep trying to defend better and improve in the final third to see if we can score a few more goals."
And speaking about the late effort from left-back Patrick van Aanholt, he said: "It was a good save as Patrick hit it very well. It looked like it was going in, especially with these balls because goalkeepers don't like them, but somehow it stayed out."
Sean Dyche was clearly delighted with his team's performance. "Sometimes I have to talk around performances but today I thought we were terrific," he said.
"The fans were excellent again and I hope they enjoyed what they saw, which was a team organised and with energy and providing the ammunition to score goals. I think, like me, they are scratching their heads a times, wondering when one is going to drop.
"One will come, but we are going about our business exactly how I planned it as a manager, which is in the right fashion that is appropriate for the team, myself and the fans. The players are delivering and there were more positive signs today. It's very, very close and it's our job now to make it happen."
He continued: "I was very pleased with the clean sheet again. It's so hard to do that back to back, but as I say the framework and energy is spot on, as was the football we played.
"I am not a zealot to just rolling it around; that's not my bag. I want the players to affect the opposition in as many ways as we can and they have the freedom to do that.
"The balance of the side was even better today. Defensively the framework of the side was good and we created enough chances to win the game. It didn't happen, but I thought overall we deserved more than we got and if we keep playing like that I am absolutely convinced that wins are around the corner.
"There were no question marks for me. I thought the performance was very good. I am not naive; I know we have to score to win games, but you also have to believe in performance levels and it's my belief that through that you win games.
"You have to create chances to win and we did that today. The one in the second half on Juke looked a penalty. Anywhere else on the pitch that's a foul, so in the box it has to be given, that's my opinion."
Dyche had praise for both Lukas Jutkiewicz and home debutant George Boyd. "I think Juke was absolutely tremendous today," he said of his striker. "I was a centre half and I wouldn't have enjoyed playing against him today. A goal would have capped things off, that's for sure.
"He has been superb for us so far and George Boyd was also excellent," and of Boyd, he said: "If there is a player who, on immediate showings, fits our culture and what we are about then it's him.
"You sign a player and you want them to fit what we are about and I felt he was fantastic today."