The Black Cats booked their trip to Wembley with a penalty shoot-out victory over reigning league champions Manchester United at Old Trafford on Wednesday night.
However, they remain entrenched in a battle for top-flight survival and after Saturday's FA Cup fourth-round clash with Kidderminster will turn their attention back to the fight for precious points.
Poyet's men have lost just one of their last 11 games in all competitions, but with only 18 points to show from their first 22 league games they have work to do to ensure the team that was good enough to see off United on their own turf have the chance to return next season.
The Uruguayan said: "I was seeing my players being brave, passing the ball, pressing, not just dropping and defending and hoping for a result.
"We were decent. We were a team that doesn't look like we are in the position we are, so we need to make sure we do it next week against Stoke at home especially and go to another level.
"If we could get three or four results in a row, then we can prepare for the final in a good manner."
Poyet had something of a reputation as a lucky player during his time at Chelsea and Tottenham, but has endured his fair share of misfortune during his short spell at the Stadium of Light to date.
However, his side survived a last-gasp United fightback and a near-comical shoot-out before securing their progression to the final.
Asked if he was a lucky manager, the 46-year-old replied: "You need to have luck, but when I played football, I wasn't just lucky, I can tell you that.
"I was not the quickest, but I was a little bit more intelligent than most of the players on the pitch. I was not the most skilful, but I have an ability to see things happening before you, so it is not like that.
"We will see as a manager - when we get to the final and at the end of the season. I believe in luck when you get it - you cannot buy it."
Sunderland face the daunting task of having to beat United's derby rivals Manchester City if they are to lift their first major trophy since 1973.
Free-scoring City will be overwhelming favourites, but the Black Cats have beaten them at home in the league in each of the last four seasons, and former Sunderland defender Gary Bennett insists the outcome will not be a foregone conclusion.
Bennett said: "They have been written off before and Manchester City will be heavy favourites, which is understandable.
"But you have to remember, on the last few occasions we have played Man City [at home], we have beaten them, so you have to bear that in mind.
"City are an excellent team, but it is on the day. Like they have shown here against Manchester United, you need a little bit of luck, you need to take take your opportunities, and you need individuals to play well, and if they do that, then they can get a result."
That would provide the perfect outcome for Salford-born full-back Phil Bardsley, whose goal against his former club looked to have spared Sunderland the penalty drama until Javier Hernandez's late strike.
The 28-year-old was discarded by Poyet's predecessor Paolo Di Canio during the summer, but once again played a key role on the sixth anniversary of his arrival at the Stadium of Light.
Bardsley told BBC Radio Newcastle: "I'm not sure I'll be allowed back into Salford, but it doesn't matter. Sunderland are going to Wembley and that's all that matters to me.
"To take this club to Wembley and to be part of such a great day out will be phenomenal. There were 9,000 there last night and they never cease to amaze me when they travel away in their thousands.
"I probably wasn't one of their popular players six months ago, but that's football. Hopefully I have turned that around and being part of this club going to Wembley is something special."