The 2-1 win on Sunday was Sunderland's first Premier League victory of the season and followed six straight defeats.
It lifted Gus Poyet's side from the foot of the table, and striker Fletcher - who scored the hosts' opener after just five minutes - says victory was critical for the club, and in particular for the fans.
Fletcher said: "We're not daft, we know what this game means to the fans and everyone around the place. We knew we needed to win this one.
"We are told every day - some fans tell us that we just need to win this one and that's it.
"The fans have not had a lot to cheer about, but I think it showed from our celebrations in the first five minutes how much we wanted to win the game.
"We managed to go on and do that. They have stuck by us and hopefully we have repaid them."
It proved a big day for Fletcher, whose impressive start to his career on Wearside has been interrupted by ankle and shoulder injuries.
He said: "That was the best feeling I've had here personally. I missed the 3-0 game last year, so that was the highlight of my Sunderland career so far.
"I got back early from my ankle and got back early from my shoulder, so it's been a good recovery. The dressing room was bouncing afterwards. The music was on and it's been a while since that was the case."
Goalkeeper Keiren Westwood knows there is still much work to be done despite the derby success.
He said: "It's a good start, we are off the bottom and (it is) onwards and upwards now. Our situation's still dire, we are not getting carried away. But we worked our socks off and we got our just rewards.
"This wasn't about being bottom of the league or conceding however many or having one point. This was about Newcastle versus Sunderland.
"It was massive, it was a cup final. We had to go out and enjoy it and have no regrets because there's been a few from the team this season."