Mowbray returns to the Sky Bet Championship after a three-month hiatus following his departure from Blackburn when his contract expired.
Neil led the Black Cats back into the second tier through the League One play-offs, but quit to take over at Stoke at the weekend.
Mowbray oversaw Sunderland's final preparations for the side's game against Rotherham on Wednesday before finalising his move to Wearside.
The 58-year-old said: "I grew up in the north east and I always remember my experiences at Roker Park with great fondness - the emotion, passion and size of Sunderland AFC.
"This is a huge opportunity to continue the good work that has taken place over the past two years and I hope the supporters can see that this football club is on its way back.
"It is a journey - I've just come from Blackburn Rovers and like here, it was a rebuilding process - but we want to win and we will be aggressive.
"The players must understand what they are fighting for because they are the standard-bearers for the city and, win, lose or draw, we must ensure those who come to watch us see a team that gives everything in every game."