The Toffees went into the game just four points adrift of fourth-placed Arsenal, but now find themselves five behind Chelsea, who have two games in hand after being leap-frogged by the Gunners.
Asked how the 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 said: "We didn't play well today. 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."
Ultimately, it was Sunderland boss Paolo Di Canio who was celebrating on the pitch after the final whistle as Verdi's "La Donna e Mobile" rang out across the Stadium of Light.
Asked about the Italian's start, rueful Moyes said: "It's a good one, isn't it? Two wins is very impressive and he will be delighted with those two results."
Di Canio was understandably thrilled with both the performance and the result - it was Sunderland's first victory over Everton in 20 attempts - which lifted his side into 14th place and above derby rivals Newcastle on goal difference.
He said: "I am very happy because it's another win and really important for the squad, for the environment. Two wins in a row - I don't know how long since it happened - to be honest, I didn't check. I don't care.
"It's important that we give consistent performances. That gives you results, gives you points in the table, so that's the most important thing at the moment."