Van Persie takes over the armband from Fabregas, who completed his protracted move back to hometown Barcelona on Monday.
Belgian defender Thomas Vermaelen has been promoted to vice-captain.
In his programme notes ahead of the Champions League qualifying play-off first leg against Udinese, he said: "Robin van Persie will be captain this season. He is a leader through the way he plays, through his attitude and through his desire to win.
"He's been at the club a long time and he's a technical leader as well. Thomas Vermaelen will assist him, it's the decision I have made and I'm confident the group will respond to it."
Van Persie, who will take the armband for this weekend's clash with Liverpool as he is suspended for the Champions League clash with Udinese, is ready to take up the responsibility that goes with being captain of the team.
"As captain you are an ambassador for the club and I'm ready for that," Van Persie told the official matchday programme.
"I've been in the game now for more than 10 years, so I know how this world works.
"When I was 18 I didn't have a clue! Every day I was always surprised by something, when I went to training or when I had a game, every day was just one big surprise for me.
"How people reacted after the game - the fans, players, media - it was all a surprise to me, I was just thinking about playing football.
"But now I know how things work and I understand that certain actions have consequences. I know that quite well, I feel experienced, so I feel ready for this.
"I know that every single player represents their club anyway, everywhere you are, 24/7. I've been dealing with it for a few years now, so I know what my part is."