Moyes has penned a four-year deal with the Black Cats.
The former Manchester United and Everton boss takes over the reins from Allardyce, who has replaced Roy Hodgson in the England hotseat.
Moyes, who won the LMA manager of the year award three times with the Toffees, said: "I am delighted to have joined Sunderland. I am relishing the challenge and excited by the opportunity.
"I have taken over a big British club, with a great support and I'm looking forward to working in the Premier League again.
"I look forward to continuing the good work done by Sam."