A miserable start to the season has seen Sunderland claim just one win from 10 league games.
However, they are now through to the last eight of the League Cup following second-half goals from Phil Bardsley and Sebastian Larsson against Southampton, who claimed a late consolation through Maya Yoshida.
Sunderland will face Chelsea in the quarter-finals on December 17, but Poyet hopes Wednesday's result has a bearing on his side's more immediate future.
"We are desperate to see the team play in a good way," the Uruguayan told BBC Sport.
"It was important for the players. It's a new way, a different way. We can do it. It's just started; it's the beginning.
"We need to make sure we are a very strong at home. You have 40,000 people behind you at home.
"We need a clean sheet now. How much we would give for a clean sheet on Sunday [in the Premier League match against Manchester City]."