Fabio Borini's penalty and a close-range effort from Adam Johnson established a 2-0 lead by half-time at St James' Park.
Jack Colback smashed home a third in the 80th minute to send the visiting supporters into raptures, and earn Sunderland a first league double over their near-neighbours since 1967.
Poyet was elated to get the win, but insisted his players deserved credit for handling the red-hot derby atmosphere.
"(It is) one of the best days I have had in football, as a manager, for sure," he said.
"The players deserve the credit, they go on the pitch and know how hard it is to play in a derby, the fans and the atmosphere, the pressure and you need to be strong but at the same time relaxed to play the football we want to play.
"The players are enjoying it and full credit to them, they were outstanding, we defended well and created lots of chances, and we scored at the right time, so it was a very good performance.
"Three points is the most important thing, but the way we did it, I don't know how many people were expecting this, but I have been saying for a long time we are a very difficult team to play against."
Poyet handed a debut to midfielder Liam Bridcutt as Lee Cattermole was still sidelined with a calf issue, and he was thrilled with the former Brighton and Hove Albion man's performance.
He added: "He knows the role. If there is one player in the world that I want to play that role it is Liam.
"Not being able to play Lee Cattermole because he has been injured it was a relief to be able to put Liam in and I had no doubt about giving him his debut.
"The best thing for him is that we won, didn't concede, he was calm and passed the ball well and kept his shape. He is a happy boy today and I am pleased for him."
