The Black Cats meet Manchester City at Wembley to contest the season's first piece of silverware, with Poyet's side entering the tie as outsiders to lift the trophy.
The former Brighton boss replaced Paolo Di Canio as Sunderland manager in October after the Italian's controversial reign was brought to an abrupt end.
Poyet has since helped turn around the Wearsiders' fortunes, taking them to the brink of a first trophy in 41 years, and he says that Short deserves to be rewarded with success for showing belief in him.
"I've said before, the chairman came into the dressing room after the semi-final at Manchester United to celebrate with us and that showed how much it means to him," he said.
"He doesn't come down too many times, but for him to go from the boardroom to the dressing room was special. We need to make him proud.
"I want to be respectful and not talk too much about him, just give him the credit he deserves. It would be nice to do it for him.
"He will be allowed in the changing room at Wembley if we win. To be honest, I'm not going to tell the chairman he's not allowed in!"
Poyet, who had never managed in the top flight prior to his appointment at the Stadium of Light, also admitted that he was grateful to Short for giving him that chance.
"But I'd like to say a few things about him. I think it was very brave of him to give me the job," he said.
"I imagine, at the time, most people were thinking of an experienced, older manager. But he took a risk in certain ways.
"I'm not going to go against myself, but he took a risk in terms of the type of person needed and the character to do certain things because of the players here and because of what had been happening. I have to thank him for that."
Sunderland are still battling to secure their Premier League status, but Poyet is hopeful of doing that as well as adding silverware to his achievements this season.
He said: "Whether it's going to be good enough, we will see, but I will try my best.
"It would mean everything to me to win the cup and keep us in the Premier League.
"I'd love to be the first Sunderland manager to win the League Cup. That would be special."
Having needed spot-kicks to see off Manchester United in the semi-finals, Poyet says that it is something the team have been thinking about, but insists that it is not what they are aiming for.
"And yes, we've practised penalties," he said. "I was hammered for not practising them before the semi-final shoot-out.
"I'd love to win on penalties - but I'm not going to play for them!''