But a draw was the least Forest deserved as they collected their first point at the City Ground this season.
"We did get into good positions, we did get down the side of Millwall's back-three. We could have done better with the final pass, we could have created more chances," said Cooper, who replaced Chris Hughton at the helm.
"But we will do that because we have ability in the team and I want to get players on the pitch who can really hurt the opposition, particularly here.
"We will try to win every game we play, that is the mentality we are going to have. But the quality will get better in time because we definitely have the ability.
"It was a tough game but one in which there were a lot of moments. If Lewis Grabban had scored his chance early on it might have been different. We had planned to stop crosses because we knew about Matt Smith and how he can make a difference when the ball is delivered in the air, so that was a disappointing goal to concede.
"But when you are losing you either accept it and get beaten or you roll your sleeves up and fight. We did not quite finish the task by winning the game but we got back in the game.
"There was an amazing atmosphere and we have a set of supporters who really want to get behind this team and we have to do our bit on the pitch. Today has really whetted my appetite for what is ahead."
Smith's towering header had put the visitors in front during a first half in which both sides had limited threat, but Forest levelled when Lowe's cross looped over the head of visiting goalkeeper Bart Bialkowski and into the net off the inside of the far post.
Millwall manager Gary Rowett felt his team could have taken all three points, with Smith also hitting the bar in the second half.
"It was a difficult game but I thought we might have won it," said Rowett. "We saw them win at Huddersfield playing this formation but we were not sure how they would do things today.
"We knew the atmosphere here is always excellent and in the first 15 minutes you have to stand up to it. On the whole we did that, aside from that one chance for Grabban.
"We controlled large parts of the first half and scored an excellent goal. It was the first time we had put a quality ball into the box, through Sheyi Ojo, and Matt is always there when you do that.
"We should have scored again before half-time and we never really looked to be in too much trouble until the end. In the first five minutes of the second half we allowed them a couple of chances, we allowed them back into the game.
"They got a massive stroke of fortune because Max Lowe clearly tried to cross the ball. You could see by his reaction afterwards that he did not mean it. It was the only place he could have put the ball, where it was possible for him to score."