On-loan Swansea City midfielder Ki Sung-Yueng netted a dramatic winner in the 118th minute of the quarter-final at the Stadium of Light on Tuesday.
Ki's strike came after Fabio Borini had struck a minute before the end of normal time for Sunderland to cancel out an own goal from Lee Cattermole.
A place in the last four comes as welcome relief for Poyet's men, who are rooted to the foot of the Premier League table.
The Uruguayan is hopeful his men can take confidence into Saturday's home game against Norwich City, having beaten Manchester City and now Chelsea on home soil.
"It's massive (the Norwich game). I said I want to win both matches, I hope we win on Saturday," Poyet told Sky Sports.
"We already beat Manchester City here, now Chelsea. It has to be the platform to win on Saturday.
"We want to maintain the consistency of players playing together in the same way and believing in what we do.
On Tuesday's victory, Poyet added: "The players have been fantastic. They did not stop attacking.
"The message at half-time was (to) keep believing. Don't change how you play even if you concede. If we stay together and start moving the ball around you create chances.
"Then extra-time you know everybody is tired and we scored."
