The Black Cats – who currently find themselves at the foot of the Premier League table – secured a shock 2-1 win after extra-time against the Blues to book their spot in the last four of the competition.
Ki Sung-Yueng netted a dramatic 118th minute winner to avoid penalty kicks, after Fabio Borini has levelled the scores late on in normal time to cancel out a Lee Cattermole own goal.
After the game Poyet praised his side for their efforts, but urged them to use the result as a springboard to turn their ailing league campaign around:
“That is massive.” He told Sky Sports.
“People ask me before, which one do you want to win? Both – I hope we can win on Saturday.”
Of whether Tuesday’s victory can be a turning point, the Uruguayan said: “I hope so.
“That was the idea we want, to maintain that consistency.
“They played fantastic, they didn’t stop.
“They gave everything today and I hope they have turned the corner.”
Sunderland are currently five points from safety with Poyet – who succeeded Paolo Di Canio earlier this season – having endured a difficult opening spell to his career at the Stadium of Light.