Al-Sadd said on their website Alves had signed a three-year contract. The fee was undisclosed.
The 28-year-old joined Boro in January 2008 from Dutch club Heerenveen and started brightly with six goals in his first 11 games before the end of his first Premier League season.
However, he only scored four times in 31 league matches last term as Middlesbrough were relegated.
He is the fourth international to leave the north-east club in recent weeks following the departures of England's Stewart Downing, Turkey's Tuncay Sanli and Germany's Robert Huth.
Boro's website said the Alves deal was complete apart from the receipt of the official paperwork.
‘I don’t think Liverpool would look at Ollie Watkins, a striker isn’t a pressing issue for them – it’s Arsenal who need one’ Former Reds star explains why his old club don’t need an out-and-out forward this summer
‘He’ll still be playing at 40 at a good level because he’s in such good shape and looks after himself so well. He does everything to be a top professional’: Ex-Liverpool coach insists Mohamed Salah has plenty more miles in the tank
