The Toffees went into the game just four points adrift of fourth-placed Arsenal but find themselves five behind Chelsea, who have two games in hand after being leap-frogged by the Gunners.
Asked how defeat on Wearside would affect their chances of closing that gap, Moyes said: "We won't get a top-four position. Every game is becoming must-win, so I can't see that happening.
"It was an important game for us and we weren't able to take any points from it."
Stephane Sessegnon's strike in first-half injury time was enough to secure a second successive Barclays Premier League victory for Sunderland, who have now eased themselves six points clear of the drop zone.
It was no more than they deserved after creating the better chances in the game with Everton struggling to test keeper Simon Mignolet.
Moyes added: "We didn't play well. We had an awful lot of pressure in the second half - we didn't have the quality, maybe, to get the goals from it, but we had an awful lot of opportunities to try to do things a little bit better."
The visitors might still have emerged with a point had referee Phil Dowd viewed Danny Rose's second-half challenge on substitute Nikica Jelavic in the same way as Moyes.
The Scot manager said: "I thought he pulled him down, yes, I thought he pulled his arm and pulled him down in the box."