Ireland international O'Shea, 30, and former England defender Brown were fringe players at Old Trafford and will hope for more regular first-team opportunities in the north-east of England.
Sunderland manager Steve Bruce, a former United player himself, said he was delighted to sign O'Shea.
"John has been a solid, consistent, part of Manchester United's squad for well over a decade which goes to show the fantastic pedigree of the player we have signed," Bruce said in a statement on the club's website.
"He is a wonderful professional both on and off the field and his wealth of experience and versatility will be great attributes for us."
Brown, 31, has spent his entire career with Manchester United, playing 361 matches for the club and winning five Premier League and two Champions League winners' medals as well as 23 England caps.
United manager Sir Alex Ferguson paid tribute to his two departing players.
"In his time with us, he [Brown] certainly became a favourite with the fans for his no-nonsense approach to the game and his infectious personality," Ferguson said in a statement on the United website.
"It's always the hardest of decisions to let loyal servants go. But life moves on and, unquestionably, Wes will get the chance to make a real contribution to Steve Bruce's Sunderland and I'm delighted to be passing him into such good hands."
Ferguson said O'Shea had been an outstanding servant.
"His versatility, calmness under pressure and his commitment to the cause at United means that fans will remember him with great affection," he said.
"John has won so many medals - and has deserved every one of them. He epitomises our club's approach as a team. After so many years of service to United, I'm pleased that John has found a career opportunity which will deliver the number of games he deserves."
United's young Irish midfielder Darron Gibson has also been linked with a move to Sunderland in newspaper reports.
