Sunderland are bottom of the Premier League with just one point in eight matches and a new boss in Gus Poyet, with Paolo di Canio having been sacked.
They take on Newcastle in the north-east derby on Sunday and Vaughan says there is no panic in the camp.
"We just need a couple of good results to get the belief there and get on a good run, like we did when Martin O'Neill came in, and we can easily climb up the table," he said.
"We need to work together. If we start blaming each other, it's only going to get worse.
"If we can stick together and if we can get a couple of wins or a few good results, then the belief is there and we can hopefully climb up the table."
O'Neill was drafted in as Steve Bruce's replacement on December 6, 2011 and immediately embarked upon a run of seven wins and a draw in his first 10 league games, and a similar return is just what is required if Poyet is to rekindle hope.
Poyet is the fourth permanent manager under whom Vaughan has worked since arriving at the Stadium of Light from Blackpool in July 2011 with Bruce, O'Neill and Paolo Di Canio having all departed in short order.
However, he admits he and his team-mates cannot afford to allow the latest upheaval to distract them.
He said: "It's just something you have got to deal with. We have probably not helped in the matter with the results that we have been able to get on the pitch.
"But we have just got to work together and stick together and try to do well for the new manager."
Sunderland and Vaughan in particular at least have fond and recent memories upon which to draw as they head into this weekend's fixture.
The Black Cats travelled to St James' Park in April as underdogs in Di Canio's second game at the helm in the midst of another survival battle, and emerged with a 3-0 victory capped by the Welshman's stunning 82nd-minute strike.
He said: "It was probably the best moment in a Sunderland shirt for me, so hopefully if I get the opportunity, I can if not score, then help the team win.
"We need something to kick-start our season, get that belief there and if we can win at the weekend against our local rivals, then everyone will be happy on Monday."