The Foxes fought back from two goals down to win as Facundo Buonanotte, Jamie Vardy (penalty) and Jordan Ayew struck for the visitors. Ryan Fraser was also sent off for the Saints.
Cooper said,"It was obviously a brilliant day and will hopefully be memorable for the season. We look at the first half and we haven't defended the goals well enough but I actually felt we were always in the game.
"Although we should have done better, I was also pretty confident that the game could change. We just asked the players to really believe in themselves and stick together and that's what we did.
"Today is a really good show of how together we are. That's what you need to have if you want to reach objectives which is what we are trying to do. It's a brilliant win, a brilliant win for the supporters, we've got to learn, continue to improve, continue to strive, and we'll continue to get better as well."
On Abdul Fatawu's impact, he stated: "We love him. He hasn't played for a couple of games now. He's a really exciting young player who still has things to learn, he's transitioning to the Premier League. We are going to give him everything to play at the level he played today. That's him taking an opportunity, showing who he is and what he can do. A brilliant example of a young player taking opportunities and performing on the stage."
On Ayew's goal, Cooper added: "It was great instinctive decision making from the players. We were a little disappointed we didn't score from set plays in the first half. We want that to be a good part of our game as well. Players coming off the bench and doing their bit as well is exactly what we want as well. I've got to look at what wasn't right in the first half as well, but it's just a good day for the supporters and players."