The Capital One Cup finalists will attempt to book a place in the FA Cup fifth round on Saturday when Skrill Premier Kidderminster head for the Stadium of Light looking to shock the men who dumped Manchester United out of the League Cup at Old Trafford on Wednesday evening.
However, 46-year-old Poyet, who arrived on Wearside in October to take charge of a club seemingly destined for the Sky Bet Championship, will not be swayed from the mission for which he was appointed by a Wembley date with Manchester City in March.
He said: "The challenge is to stay in the Premier League. That was my challenge when I got the job and it was clear from the beginning.
"Now we are lucky enough to find ourselves in a great position in a cup competition to go and play a final, and we need to try to do everything possible to win it.
"People are talking to me about 'the treble' - staying up and winning the two cups.
"Now it's a new treble. Normally it's winning everything, but staying up has become like a trophy.
"I am realistic - I am realistic in the [League] cup as well because the opposition is not an easy one. Right now, they are on fire and anything can happen, so we need to be really strong on the day."
Wearside has been engulfed by cup frenzy since the remarkable penalty shoot-out which heaped more pressure on United boss David Moyes and left his opposite number wreathed in smiles.
Poyet has spent much of his time at Sunderland to date fighting to drag the Black Cats to within touching distance of league safety, and admits that leading his team out at Wembley did not figure even in his wildest dreams.
He said: "No. I am realistic, no. I wasn't even thinking about the cup, to be honest, I was thinking just to win a football game. That was the aim first.
"Then after winning a couple of games, the idea was to be consistent, to be more organised and difficult to beat, to start playing a certain style of football and to start getting better slowly.
"But I don't think you can plan to go ahead and dream of a cup final, not at all."
In the short-term, Poyet's task will be to put together a team - he will make wholesale changes but stop short of fielding an entirely different XI - which will have to negotiate the switch from midweek underdogs to overwhelming weekend favourites.
He said: "It's the most important part of the job yesterday, today and tomorrow, the mental side. We have to make sure, us as a staff, that we put the players in there realising what can happen and make sure that it doesn't.
"I am sure they [Kidderminster] are going to be a threat, I can tell you that, and I am sure they are going to make it very difficult for us to break them down."
Keeper Oscar Ustari and central defender Santiago Vergini will both be handed debuts just days after signing for the club.
Meanwhile, Poyet has confirmed that no bid for midfielder Lee Cattermole has been received despite persistent speculation that he is a target for Stoke, who are due at the Stadium of Light on Wednesday evening.
He said: "There haven't been any offers. It's very easy for me to say that.
"It's difficult because the team that is being linked is the team that we are playing next week, so it's not normal, I would say. It's a tricky situation, I suppose for them as well.
"If they really want him, whether they do it before or after, we will see."