Substitute Fabio Borini's 85th-minute pile-driver secured a 2-1 win at the Stadium of Light, the Black Cats' first in the league this season, and lifted them from the foot of the table.
However, Poyet's men remain six points adrift of 17th place despite ending a run of six successive defeats, and the Uruguayan is looking for more of the same after matching predecessor Paolo Di Canio's feat of beating the Magpies in just his second game at the helm.
The 45-year-old said: "Look, I didn't like to see the table. When you are bottom, it's not nice. We are off the bottom - that's something; we beat the biggest rivals - that's even bigger.
"Now we need to keep going. We need to start passing the ball better, we need to keep believing and we need to do different things.
"This is what we needed so, from now on, there are no excuses because we needed a lift and there is no better lift than this one, so I am very happy for the players and absolutely delighted for the fans.
"I saw what it means all this week in town. I imagine how it is going to be this week - I don't know how much it matters being second from bottom, it's just about beating the rivals and everyone will have a nice week, I will tell you that."
Borini, a 69th-minute replacement for Adam Johnson, won the game in thrilling style when he smashed an unstoppable shot past Tim Krul from Jozy Altidore's lay-off to finally kick-start Sunderland's season.
His decisive goal came with Newcastle looking the more likely victors after Mathieu Debuchy had cancelled out Steven Fletcher's fifth-minute opener 12 minutes after the restart.
But with derby specialist Shola Ameobi, who has seven goals against the Black Cats to his name, having been thrown into the fray, it was fellow replacement Borini who stole the headlines.
Poyet said: "Fabio was so close to playing, the way he has been training, his attitude, how positive he is. But sometimes you can't play for whatever reasons, because of the shape, because of the team, because of the time.
"If someone deserved to score today, it was Fabio Borini."
The win further enhanced Poyet's reputation as the scourge of Newcastle and extended his 100 per cent managerial record against the Magpies to three games.
He said with a smile: "You know what, they asked me before the game and I thought 'it's not going to change now when I need three points'.
"I don't know why, but I am sure they are going to hate me even more. I don't know why or how it happens. Sometimes there is no explanation."
Opposite number Alan Pardew, however, was convinced his side should have emerged victorious, and was unhappy with a free-kick awarded by referee Lee Probert for Cheick Tiote's challenge on Fletcher in the lead-up to the goal.
Pardew, who is the first Newcastle manager to lose back-to-back derbies since 1966-67, said: "When we got the equaliser, the more the game went on the more control we got.
"We had two or three good chances, I thought, at 1-1 and if we got the second goal, the game was over because they would have been dead.
"Unfortunately for us, we didn't get that second goal. Unfortunately, the referee played a part in this defeat as well because that was definitely not a free-kick in my opinion.
"The linesman gave us a throw, the ref gave a foul and I think one or two of ours switched off.
"They certainly didn't think there was a foul, the boys, and then they just slipped through us and the one chance they had second half, he smashed it in, so it's a really tough one for us to take."
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