The defeat at the Stadium of Light leaves the Toffees four points behind Chelsea in fourth in the Premier League, with the Merseyside club having played two more games than their rivals.
Manager Moyes told BBC Sport: "We probably didn't deserve to lose the game but we didn't play well enough to win it. It was a poor game. I don't think now we can make the Champions League.
"Mathematically we still can, but I think it's unlikely the teams above us are going to drop points. What we'll do is see if we can reach the Europa League and keep pushing for that. We do our best to compete."
The loss to Sunderland was Everton's first defeat for six games.