The striker has signed a three-and-a-half-year deal at the Stadium of Light, just days after he was booed by some of the Black Cats fans when he appeared against them for the Swans.
"As soon as I heard about the interest from Sunderland I didn't even have to consider any of the other options that were on the table," the 27-year-old told safc.com.
"I was born and brought up in the north east so I know first-hand the intense passion that people have for their football in this part of the world - it's special and there's nowhere else like it."
Graham has been manager Martin O'Neill's top transfer target this month and the Ulsterman has had to remain patient with his club's bid on the table for some time before Swansea eventually decided to accept it.
However, they had been confident all along that Graham was keen to move to the Stadium of Light and have held their nerve to get their man.
His initial task will be to persuade those supporters who jeered his introduction during the second half of Tuesday night's 0-0 draw at the Stadium of Light that his Newcastle roots and affections will be no bar to success on Wearside.
Graham said: "Of course, I totally understand that some fans might need convincing about my commitment to Sunderland because of my roots, but I've always given 100 per cent no matter what team I play for and no matter who I play against.
"It's up to me to show people I deserve to wear the Sunderland shirt, and I'm excited about playing for a club as big as this.
"I'm really excited about the future - this is a new phase of my career and I can't wait to get started."
The reaction on Tuesday night prompted former club Middlesbrough to make their own enquiry about Graham, although it soon emerged that he was intent on remaining in the Barclays Premier League with Sunderland.
Graham will provide much-needed back-up to 10-goal leading scorer Steven Fletcher following the departures of Fraizer Campbell and Louis Saha and Ji Dong-won's loan move to German club Augsburg.
Connor Wickham is the only other senior striker on the books, although James McFadden could win an extended stay after showing signs of rediscovering his form during an initial three-month stay.