The Irish U21 international has linked up with the former Irish boss and City defender in a 3-year deal, with the Blues picking up a nominal fee.
"I had various options but when I heard about Sunderland's interest, I was thrilled," Elliott told BBC Sport.
The 20-year-old starred for the City reserves last season, and also at the Youth World Cup in December, and was rewarded with his Premiership debut as a substitute at Bolton in February.
However, with his City contract due to expire, boss Kevin Keegan has decided to let the youngster move on.