The Toffees went into the game just four points adrift of fourth-placed Arsenal, but tonight 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 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."
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."
But ultimately, it was 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 win 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."
However, Di Canio, who described his emotions during and after his first home game as among the best of his 27-year career in football, refused to concede that one more victory would be enough to secure the club's top-flight status.
He said: "I don't think it's enough. Every weekend, our prediction changes. If Wigan can win against Swansea, they are 34, three points from the main group close to them on 37 with four games to go.
"I don't know, it depends. If you get three points against Aston Villa, it can probably be enough, 40, 41; if you lose against Aston Villa or Stoke and then you beat Southampton, maybe it's not enough.
"I don't know. If we win the direct conflict (with Villa), 40, 41 can be enough, otherwise it's difficult."
The 44-year-old Italian was a little more reserved on the sidelines than he was at Newcastle last weekend, where he slid along the turf on his knees as Adam Johnson put the Black Cats 2-0 up.
There was no repeat on Saturday, and Di Canio later revealed the reason why.
He said: "It was too early. I can't dive every time. Already yesterday, it was difficult to find another pair of trousers.
"The tailor came and I wasn't happy at the beginning because they were too short and too soft.
"I said: 'You have to make sure I have the right ones for tomorrow', otherwise I was going to wear the ones I used last time dirty."