The Black Cats boss, while delighted with their 'best performance away from home', asked for more consistency following his side's goalless draw at Villa Park
The 46-year-old guided his side to their second point on the road this season but saw Giaccherini slice the ball over the bar from just two yards out after Phil Bardsley had headed the ball accross goal.
But Poyet insists that, despite the player feeling upset by the incident, he is not too downbeat and instead focused his energy on the strong performance of his side, urging them to re-create it in future games.
"He’s so sad now," he said, when asked about Giaccherini's miss on goal. "He doesn’t stop saying sorry. It’s football. Things happen for a reason sometimes and the main thing is to perform for 90 minutes.
"For us it was, without any doubt, out best performance away from home. Why? We made it difficult for Villa, we were brave on the ball and we created two or three very good chances.
"For a team down at the bottom and away from home, I don’t think it is too bad. Maybe we deserved the three points today. But I want this team. I want that team on the pitch every week."
Sunderland saw the point push them off the bottom of the table and although Paul Lambert was disappointed his side couldn't claim the win, he was pleased to extend their unbeaten run to four games.
"I don’t think we deserved to win the game," admitted Lambert in his press-conference. "We didn’t do enough to win the game. I don’t think we were going to lose it but I don’t think we were going to win it.
"That’s four unbeaten and you’ve always got to look to the positive. It was a game where we might have lost it last year and that’s a good thing.
"It’s another clean sheet, it’s another point and it’s another game unbeaten. I’m disappointed not to win the game but you’ve always got to have a positive side of it."