The former Man United manager’s side have managed just five victories this term and with four games left to play there was no way to escape.
The Black Cats will now drop down to the Championship, 10 years after they won promotion to the top-flight under former boss Roy Keane.
Joshua King condemned Sunderland to the drop with an 88th minute goal as the Cherries took all three points from their trip to the Stadium of Light.
Speaking after the the game, Moyes told Sky Sports reporters: “I would like to bounce back with them.
“The job is to make sure we get prepared and ready, but it’s too quick to talk about it.”
When asked whether he sees himself in charge of the club next season, he replied: “I would like to make sure we get Sunderland back in the Premier League.”
The home fans sang: “We are going down, we are going down,” as their side spluttered to another home defeat on Saturday.
Moyes suffered chants for him to leave the hot-seat, in a miserable end to their slim hopes of staying up.
The club are now doomed to financial crisis and a season on vastly reduced TV payments in the Champinship.
They are also set to post one of the worst six seasons in top flight history.
Sunderland have flirted with the drop for years, and now it has come.