Advocaat revels in 'deserved' derby win
Jermain Defoe's stunning volley just before the break moved Sunderland into 15th place and three points clear of the Premier League relegation zone.
Advocaat felt Sunderland could have won by a greater margin, but urged his players not to get carried away as they bid to secure their top-flight status.
"It was working from beginning to end. We were tired at the end and they start playing high ball but it was a well-deserved win," the Dutchman told Sky Sports.
"I'm really proud of them. Besides the goal, which was brilliant, we still had good opportunities.
"We gave one chance away but overall we really deserved it. I said it before, if you can get fans behind you that gives you an extra boost.
"We know we have three home games to come, we have to get points there, but from tomorrow we will start concentrating on Crystal Palace."
Defoe made his finest contribution for Sunderland since his January move from Toronto FC when he fired home Steven Fletcher's knockdown on the stroke of half-time.
The striker said: "It's one of my best [goals], with my left foot as well. In a game like this, even if it's a tap-in it's massive to get three points.
"To score a goal like that is amazing. As soon as it leaves your feet, it's an amazing feeling.
"It was quite emotional really, I just wanted the three points so badly.
"I tried to blank [the pressure] out, sometimes when you think about pressure you get tense and maybe don't perform.
"But if we play like that we'll be fine."