Sunderland fans turned against head coach Poyet during last Sunday's surprise 2-0 FA Cup fifth round defeat at League One side Bradford City.
The Uruguayan hit out at the media after his side's shock exit at Valley Parade and on Friday made public a plea to fans via the club's official website.
He wrote: "What we achieved last year was achieved by all of us together: club (chairman, directors, staff and players) and fans.
"Nobody, and I mean nobody, from outside helped us a bit, it was only us together.
"During my career as a player, coach or manager, I have always had a fantastic relationship with the fans.
"From my time in Uruguay going through every team in Europe, always the relationship was strong, one of mutual respect and understanding.
"I showed from inside or from outside the pitch, my commitment, passion and dedication in achieving what every club deserved.
"So let's make sure that we don't let anyone to break our relationship, I promise you that I care and want to win as much as you do and no one thinks more time every day about the team than I do.
"I invite all of you to stay positive, to be strong, closer to each other and keep believing in what we started together last year, working harder to make it better this season.
"To finish I would like to clarify once more: the main [man] responsible [for] the results of our team is me, I always said it and I will always accept my responsibility.
"I am really looking forward to see the Stadium of Light packed on Saturday, pushing the team to what could be a very important victory."
