In his second game in charge, Poyet's Black Cats took the lead via Steven Fletcher before Mathieu Debuchy equalized midway through the second half, but Fabio Borini lashed in a winner to give the home side a 2-1 victory.
The Uruguayan manager was pleased to pick up a first win of the league campaign in the Tyne-Wear derby, but demanded that his players do not rest on their laurels.
"We beat our biggest rivals," Poyet told reporters after the game. "Now we need to keep going, now we need to start passing the ball better, now we need to keep believing.
"From now on there are no excuses. We needed a lift, well there is no bigger lift than this one. I'm very happy for our players and I'm absolutely delighted for our fans."
Meanwhile, Alan Pardew was despondent following the defeat and felt that his team was better than Sunderland overall despite having nothing to show for its efforts at the Stadium of Light.
"We had two or three good chances, I thought, at 1-1, and if we'd got that second goal the game would have been over, because they would have been dead," he said. "The only chance in the second half and he [Borini] smashes it in. It was a real tough one to take.
"I thought we played well from about 20 minutes in - right up until the second goal I was fairly pleased. I feel really disappointed for our fans and for the team. I didn't they deserved that. They just switched off and a little bit of unprofessionalism cost us."