The Netherlands international struggled to break into the first team at Stamford Bridge, making just two league appearances in five years at the club.
After loan spells with Coventry City, Newcastle United, Leicester City and Wigan Athletic, Van Aanholt spent the last two and a half seasons at Vitesse in the Eredivisie.
On Friday, it was confirmed that the 23-year-old had joined Sunderland on a four-year deal, and Van Aanholt is thrilled at the opportunity to display his qualities.
"I thought about it for a second and said, 'Yes I would love to come to Sunderland'," he told the club's official website.
"It's a great club and a great opportunity for me to play in the Premier League.
"It's a good opportunity for me after two and a half years in Holland and a couple more loans in the Championship.
"I want to prove myself in the Premier League so everyone can see what kind of player I am and what I'm capable of doing.
"I had a couple more offers from different coaches but I said no I want to stay in the Premier League."