The Black Cats' 1-0 win at Goodison Park was only the third time they had taken maximum points from a game all season and ahead of Saturday's trip to managerless Cardiff, the Wearside outfit are bottom of the table, three points adrift of safety.
After the match in Wales, Sunderland's league schedule continues with four fixtures in January - a home clash with Aston Villa, an away encounter with Fulham and then visits to the Stadium of Light by Southampton and Stoke.
The goal that steered them to the win over fifth-placed Everton came about when Ki Sung-yueng looked to take advantage of a mix-up between Toffees goalkeeper Tim Howard and Leon Osman, hit the turf in the box when challenged by the former - who was sent off - and converted the resulting penalty in the 25th minute.
And asked if he thought that mistake by Everton might prove to be the point at which the campaign turned for his side, Poyet said: "I hope. I have answered that question a few times, but I hope we can stay up.
"What I know is that I want to win a couple of games before the end of January - at least two.
"We drew (the game prior to the Everton match, 0-0 at home against Norwich) and we then we won.
"Let's see if we can win again, be happier with the points and then reorganise after the January transfer window."
Before the Norwich fixture, which extended Sunderland's winless run in the league to six games, Poyet said his players would have "no excuses" to not perform against the Canaries given the confidence they should have taken from their previous result, a 2-1 extra-time Capital One Cup quarter-final victory over Chelsea.
And it is a message the Uruguayan has reiterated off the back of them taking another considerable scalp in Everton.
Asked if that win - the Black Cats' first away from home this season - would give the team greater belief going into the clash with 16th-placed Cardiff, Poyet said: "It has to, and there can be no more excuses.
"I don't want 'the flight was not on time', 'the bus was whenever' or 'there was a party at the hotel'. I want none of that.
"I want to put a team on the pitch without excuses, and on Saturday - apart from a few heavy legs, because we worked hard (against Everton) - there are no excuses.
"Confidence should be high."
While asserting that his team should be feeling good for the Cardiff test, Poyet has also once more made it clear he is unhappy about the hectic nature of Sunderland's schedule.
Returning to another topic he had discussed ahead of the Norwich game, Poyet said of the trip to the Cardiff City Stadium: "What is going to happen on Saturday? Who knows?
"I don't know who organised our fixtures, to make us come to Everton on Boxing Day and then go to Cardiff - the enemy probably organised it for us.
"Why are we not playing on Sunday or Monday? It has to be the 28th? Apparently, yes."
Sunderland will again be without Wes Brown for Saturday's 5.30pm kick-off, while it remains to be seen whether or not his fellow defender John O'Shea is able to feature.
The pair missed the Everton win, with Brown starting a three-match suspension and O'Shea having been nursing a shoulder problem.
Definitely still ruled out through injury are goalkeeper Keiren Westwood (shoulder) and yet another defender, Carlos Cuellar (hip).