The 23-year-old Wales international has put pen to paper on a four-year deal with the Black Cats which ties him to the club until 2019.
Sunderland head coach Dick Advocaat told the club’s official website: “We know that we need quality players to improve competition for places in the side, so Adam is a welcome addition to our squad and we are very pleased to welcome him to Sunderland.”
The club’s sporting director Lee Congerton added: “Adam is a player we have been tracking for some time.
“Even though he is only 23 years old, he has a wealth of experience, and I am sure he will continue to improve under the leadership of Dick Advocaat.”
Right-back Matthews joined Celtic from Cardiff City in February 2011 and has played over 100 games for the Scottish Premiership champions, winning four titles during his time at Parkhead.
He has also represented Wales from U17 level up to the senior side where he has played for 12 times.
Celtic manager Ronny Deila reckoned it was an easy decision to sell Matthews.
“One year left on his contract, a good offer, and he wanted to try something else,” he said. “It was the right time to do it.”
Advocaat is currently rebuilding his Sunderland squad and this week signed Liverpool defender Sebastien Coates on a four-year deal as well as securing the future of captain Lee Cattermole who signed a new five-year deal.
The Black Cats are also in talks with Zenit Saint Petersburg and Dynamo Kiev over the respective signings of Nicolas Lombaerts and Jeremain Lens, according to Sky sources.