After losing skipper John O’Shea to injury before half time, two goals from Stuani gave the visitors the led at the break both assisted by Alvaro Negredo.
It was very promising start for the Black Cats as they had the appetite for goal as they closed down quickly, but a Stuani stunned the Stadium of Light to silence with a 13th minute rocket 25-yards out. It was a world-class goal and Vito Mannone couldn't do anything to keep out the 100th league goal he has conceded.
Despite a late Patrick van Aanholt goal which handed Sunderland a lifeline, a spirited second half display was not enough to the Black Cats from the game as Middlesbrough return to Teeside with all three points in the first derby since 2006.
Moyes made three changes to the starting line-up, with his hand forced due to Younes Kaboul’s recent departure and a list of injuries.
Game of two halves
The on-loan Manchester United player Adnan Januzaj sparked an impressive debut in a red and white strip and consistently beat his opposite number on every occasion but it wasn’t enough as Sunderland were behind after 13 minutes.
Stuani’s stunning goal handed the Boro the unlikely advantage at the Stadium of Light, despite having such a little attacking threat.
The hosts were unable to make the comeback, as they were dealt another blow before the break as O’Shea became the latest to join the injury list, which forced Moyes into an early substitution.
Boro certainly took full advantage of the situation as Negredo unselfishly picked out Stuani, who on the stroke of half-time, doubled Boro's lead to 2-0.
The goal on the stroke of half-time left a mountain to climb for the Black Cats, who made a change at the break with Jeremain Lens replacing Paddy McNair in the middle.
Sunderland desperately wanted to get back into the game but it just wasn’t happening for them, as they needed that little spark to ignite the squad as they went a little flat in the opening of the second half.
This soon changed as Moyes' side managed to keep the ball a lot better compared to the first as they throw everything they had at Middlesbrough, but as the chances racked up for the Black Cats they couldn’t find the back of the net.
Sunderland put themselves back in the game in the 71 minute as the Boro backline were finally breached as Ducan Watmore's initial powerful drive blocked, it was pushed into van Aanholt's path as he followed in to slot the ball into the back of the net. They gave themselves a lifeline but they couldn’t get too carried away as they need to keep defending.
Within minutes the Black Cats could have levelled the score as Guzan produced a double-save to stop Januzaj and Donald Love. As Boro held firm to condemn Sunderland to their second straight defeat.