Poyet made nine changes to his team on Saturday, and saw them move into the last eight thanks to a stunning long-range strike from Craig Gardner five minutes after the break at the Stadium of Light.
Having ended Southampton's seven-match unbeaten run in all competitions to secure their progression, Poyet said he was pleased with the spirit Sunderland showed.
"We have a got a good way of playing now and fighting for the club," the Uruguayan told BT Sport.
"Sometimes you need the whole group and sometimes when these games go for you it is important for the rest of the season."
Sunderland captain John O'Shea, who kept his place in the starting XI, revealed that Poyet had told the lesser used members of his squad to maximise their opportunity to shine.
"Gus Poyet said to the lads beforehand, make a statement," the defender explained.
"We have big games coming up, Craig Gardner's goal was fantastic and he does do it every day in training.
"Southampton had some very dangerous players and they have some quality but we had to be alive at the back and maybe the pitch helped us out a bit."
‘The pure simplicity of the way Slot has managed the squad is probably the biggest thing I could say about him. It’s not broken, so let’s get on with it’: Liverpool legend full of admiration for Jurgen Klopp's successor at Anfield
‘I trained at Spurs and thought they’d give me a chance. But I received a letter thanking me but saying they didn’t think I was good enough – I was gutted’: How Tottenham missed out on signing Wales legend John Toshack
