It was also the first win under Allardyce, who replaced Dutchman Dick Advocaat just eight games into the season.
"The boys have got a lot of confidence now with the new manager coming in," Fletcher was quoted as saying by the British media.
"He's a lot more organised. He is probably the most organised manager I have worked with. He demands a lot from everyone but he works hard on the training pitch as well, which is good too.
"Sam Allardyce is a big man with a big presence. When he walks in the room everyone shuts up and listens to him. So that's a good start," the Scotland international added.
Sunderland have now made it six consecutive derby wins over Newcastle and Fletcher hopes that the win will provide a platform for them to build on as they try to get out of the relegation zone.
"We hope it (beating Newcastle) can kickstart our season. We need to grow on this now and take it into the next few games," Fletcher said.
"The gaffer has got us organised and is trying to make us hard to beat. We need to stop leaking goals and we need to score more. We did that against Newcastle."
Sunderland's last four derby wins have been followed by defeats and goalkeeper Costel Pantilimon said he is keen to end that run when they travel to face an out-of-sorts Everton on Sunday.
"It is important to continue our better form. One win is not enough. We have to continue like this," Pantilimon said.
"We have to learn from the past (losing after derbies) to be more focused and concentrate on the next game. It is important for the group to have a good run now," the former Manchester City goalkeeper added.
(Reporting by Shravanth Vijayakumar in Bengaluru; editing by Sudipto Ganguly)
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