The Black Cats are rock bottom of the Premier League, five points adrift from safety, but have found wins easier to come by in the cup, beating League One sides MK Dons and Peterborough before seeing off Southampton to set up a meeting the the Blues.
And Paolo Di Canio's successor sees any match as an opportunity to better their Premier League form, even if cup games are seen as less important nowadays.
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"It's important. It's a cup competition. We are losing a little bit of belief in England in the cups because, of course, the Premier League or the Champions League means everything.
"It's a competition and you cannot start the season thinking: 'I won't care about this one, I will care a little bit about this one and only care properly about this one'.
"It's just about playing and becoming winners. A winner is somebody who wins most of the time and that's what we are trying to achieve here."
After an initial boost in form, Poyet has struggled to lift the Black Cats out of the quagmire in his two months in charge but insists that he is not worried about his future in the wake of the exits of fellow managers Steve Clarke and Andre Villas-Boas.
"I never think about it," he declared. "I just try to do my job and if, one day, you get a phone call and you need to go to see the chairman, then you will think: 'Oh, here we go'. But at the moment, I am just thinking about doing the job."