Coote said: "My name will be in the newspapers for the goal, but Graham (Barrett) should take most of the praise. It's my job to score goals.
"It was a terrific cross from Graham. It was a great ball in. I saw it in the air and the defender was caught out. The next thing the ball was in the back of the net.
"I suppose it was a classic goalscorer's goal. We have been practising this on the training ground, getting crosses into the box.
"Graham deserves to take a lot of the credit. I need that sort of service. It was a special moment for me. It's always very special to score your first goal for a new club.
"I was a bit confused when the ball hit the net. In previous games, one half of the stand behind the goal had been filled with Colchester fans, so I went over to them.
"Yet Brentford must have brought a lot of supporters themselves, because they had both halves of that end. I put my arms up to celebrate, only to be greeted by a sea of red and black shirts," added Coote.
Coote commented on his booking.
"One more booking and I'll be suspended, so I have to be careful, especially as I've just got myself into the side," continued Coote.
"I have taken time to adapt to the pace of the game, after spending so much time playing reserve team football, but at least two of the bookings have been very silly.
"I'm not a dirty player, which makes it all the more frustrating.
"I was booked for dissent. A defender was clearly tugging me back a couple of times, and I just ran after the referee to ask him why he hadn't given a free kick. I had to shout because of the noise, but he then booked me for chasing after him.
"Yet this was a good result for us. It's stopped the run of defeats, and hopefully there are better things to come. We've stood up and been counted today, against a very physical side," added Coote.
"It was a terrific cross from Graham. It was a great ball in. I saw it in the air and the defender was caught out. The next thing the ball was in the back of the net.
"I suppose it was a classic goalscorer's goal. We have been practising this on the training ground, getting crosses into the box.
"Graham deserves to take a lot of the credit. I need that sort of service. It was a special moment for me. It's always very special to score your first goal for a new club.
"I was a bit confused when the ball hit the net. In previous games, one half of the stand behind the goal had been filled with Colchester fans, so I went over to them.
"Yet Brentford must have brought a lot of supporters themselves, because they had both halves of that end. I put my arms up to celebrate, only to be greeted by a sea of red and black shirts," added Coote.
Coote commented on his booking.
"One more booking and I'll be suspended, so I have to be careful, especially as I've just got myself into the side," continued Coote.
"I have taken time to adapt to the pace of the game, after spending so much time playing reserve team football, but at least two of the bookings have been very silly.
"I'm not a dirty player, which makes it all the more frustrating.
"I was booked for dissent. A defender was clearly tugging me back a couple of times, and I just ran after the referee to ask him why he hadn't given a free kick. I had to shout because of the noise, but he then booked me for chasing after him.
"Yet this was a good result for us. It's stopped the run of defeats, and hopefully there are better things to come. We've stood up and been counted today, against a very physical side," added Coote.
Meanwhile Colchester United manager said "We played very well against Brentford and it’s our best performance over ninety minutes for some time. We played well for an hour against Swindon but managed it for ninety against Brentford and probably deserved to win if the truth be known."
"It’s nice for Adrian to score, he’s been patient for his chance and waited in the wings, but he’s taken his opportunity and scored a good goal today. Although we had three strikers out today, we’re never any different – regardless of what’s happening, we try to remain positive. I’ve said all along, if you play the way we play there’s going to be times where you get bad results, but we’ll never be negative. We’ll continue to play the way we know we can play and the way that suits us best."
"Hard work and doing the simple things right breeds confidence and I think they will have gained a lot of confidence out of today’s result. Especially when they went one down you wonder how we’re going to react, but every one of them was tremendous and the spirit and character shone through."
"The goal against could again have been avoided, three of their players touched the ball from the free-kick without any of our lads getting close to it. From that point of view it was disappointing, but I’m very, very pleased with how we came back into the game and deserved a point or maybe more from the second half performance."
"The goal lifted us, but it was important that we didn’t go cavalier, I was screaming at the players to keep their discipline and their shape, which is where we’ve been caught in recent weeks. Brentford are another big strong side and you’ve got to deal with them – when we do win the ball back, it’s important to play our way, which we did today."