Forest found a late equaliser through Ramon Sosa to earn the point. Forest boss Nuno and Brighton counterpart Fabian Hurzeler both saw red after Morgan Gibbs-White was sent off for a tackle on Seagulls striker Joao Pedro.
Yates later said: "Very happy. We know how good Brighton are, from back to front they have quality all over the pitch. They have a fantastic young manager and play the game the right way.
"We created some dangerous attacks and they dominated for the most part but we created our chances. So we take the point.
"We caught them by surprise - we had a game plan to get down the sides and Callum did well to win the penalty. But with good teams if you don't stop them building back into it and gaining composure or stop their home crowd getting up then it can quickly be two.
"Half-time came at a good time, we re-grouped and we just wanted to show the fight and resilience that we have done all season."
On the red cards, he said: "Nothing too much has been said really. We leave that down to the referee, that is his call and what he is paid for. So yeah nothing much to say on it."