The 26-year-old defender initially moved to the Premier League side in January for the remainder of the 2013/14 campaign, and re-signed for the Black Cats last month for this season.
The negotiations between Sunderland and Estudiantes were long and Vergini is relieved that he could remain part of Gus Poyet's team.
"Yes, my initial thought was to have come back over before then (August), but it didn't work out like that," he told the Sunderland Echo.
"So, yes, there was a fair deal of suspense from it all and, yes, the wait really was quite a long one. But, in the end, the outcome has been a positive one and the one we all wanted."
Sunderland will return to Premier League action on Saturday when they host Tottenham Hotspur at the Stadium of Light.
Last season the Black Cats were fighting against relegation, but they have made a good start to the 2014/15 campaign with two points in three matches, including a 1-1 draw with Manchester United at home.
"The context now is really very different to back then," said Vergini. "Then the situation was very difficult and we did need to win that particular game.
"Now we go with renewed hope and the context is very different. We hope to do a lot better. Of course, we're playing at home and we'll have the support of our own fans, which is very important.
"I have to say that I quite like the way that we've been working as a team in these first few games, but it's absolutely vital to get points on the board."