O'Shea, 30, travelled to Wearside for a medical on Thursday morning and wasted little time in becoming Steve Bruce's eighth summer signing.
Like Brown, he has joined for an undisclosed fee after United last month accepted a joint offer of more than £12million for the pair and midfielder Darron Gibson.
O'Shea said: "Once I'd heard reports about the club and the people here, the ambitions and what the manager was all about, it was the only place I was going to come to."
Manager Steve Bruce added: "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 player we have signed.
"He is a wonderful professional both on and off the field and his wealth of experience and versatility will be great attributes for us.
"I'm delighted with the players we have been able to secure ahead of the new season."
O'Shea boasts five Premier League titles, an FA Cup winners' medal, three League Cup successes and a Champions League triumph among his list of achievements at Old Trafford.
He is also a stalwart of Giovanni Trapattoni's Ireland team and like Brown, will bring experience and a winning mentality to the Stadium of Light.
He made almost 400 appearances for the Red Devils during his 12 years with the club and it was initially thought he would take some persuading to leave.
However, despite having to delay talks earlier this week as his wife gave birth, he managed to tie up his move swiftly on Thursday.
Brown, 31, managed barely 30 of his 361 appearances for United in the last three Premier League campaigns with injuries having taken their toll.
That is a statistic he is determined to address after agreeing to move to Wearside.
Asked about his ambitions, he told SAFC TV: "Last season, I was fine, I just didn't get into the team, so it was a little bit frustrating. But that's football and you move on.
"I am just delighted to be here now to help Sunderland push forward.
"Personally, it would be good to be playing every week, trying to help the team win and playing well.
"Club-wise, everyone will be looking to push the club forward and start challenging for those European places - and who knows beyond there?
"That's what teams always aim for at the beginning of the season, and we will be no different.
"But obviously, the squad will be a lot stronger."
Brown admitted that the hunger to be involved week in, week out made his decision to leave United reasonably straightforward, but with several options having been presented to him, Bruce was a major factor.
He said: "He's been great over the last few weeks when I have been talking to him.
"Straight away, I said it would be great if I could join and we could get it all sorted, and we have done and I am here.
"When I was younger, I used to watch Steve and I have known him a little bit as well, so it was fine."
Bruce, who launched his joint bid for the United trio as Jordan Henderson headed for Liverpool last month, was thrilled to have signed the defender.
He said: "We are delighted to welcome a player of Wes' ability, experience and character to the club.
"He has been a fantastic professional for Manchester United through the years and not only will he strengthen our defensive options, he will also bring with him a winning mentality and level of maturity which can only be positive for the players around him."
Meanwhile, the Black Cats were tonight also closing in on a free transfer move for Blackpool midfielder David Vaughan, who is understood to have travelled across the Pennines earlier on Thursday with the intention of sealing his own switch.
In a busy day for the Wearside club, legal director Margaret Byrne was appointed new chief executive and Bryan "Pop" Robson returned as chief scout.