The Capital One Cup finalists marched into the FA Cup quarter-finals with a 1-0 victory over Southampton to leave their fans dreaming of a second opportunity to end the club's 41-year wait for silverware.
However, they remain entrenched in a relegation fight and that remains Poyet's overwhelming priority.
Reminded that Bob Stokoe, who led the Black Cats to FA Cup glory in 1973, is commemorated by a statue at the Stadium of Light, the 46-year-old said with a smile: "I would prefer to stay in the Premier League.
"Apparently to be more famous, it's better to win the cup. I don't want any more people knowing me, I think they know me enough, so I would prefer to stay in the Premier League.
"I am not looking for a statue, that's for sure.
"It's a personal feeling. I can't answer for anybody else, it's me. I would be devastated if we go down, so it's me.
"Now at the same time, I have to say if we win the cup, I am going to be jumping up and down and I am going to be wanting to come back here and celebrate with the fans, that's true.
"We will see, we will see what happens."
Sunderland face Manchester City in the Capital One Cup final on March 2 and could be just 90 minutes away from a second trip to Wembley, which will host the FA Cup semi-finals.
However, Poyet admits cup success was the last thing on his mind when he replaced Paolo Di Canio at the helm in October.
He said: "Look, the day we came here at 11.45pm when we arrived for the first time at the training ground, there was only one thing on my mind, the Barclays Premier League.
"I need to be honest, I didn't even think about the cup at that time.
"I don't know - are we a cup team? It looks like it. We are going to play it. Let's see who the draw brings and who knows what is going to happen?
"But it is working for us, so we will keep going."
A game of little genuine quality was settled four minutes into the second half when midfielder Craig Gardner latched on to a loose ball 22 yards out and smashed it past keeper Kelvin Davis off the underside of the crossbar.
However, Southampton, who were below-par throughout, could have snatched a draw at the death with Adam Lallana and Rickie Lambert both failing to make the most of driven Nathaniel Clyne crosses, with Lambert firing over an open goal from just three yards.
Poyet made nine changes to the team which lost to Hull last weekend but saw his understudies get the better of a Southampton side which included only five of the men who started against the Tigers on Tuesday evening.
Assistant manager Jesus Perez, standing in for Mauricio Pochettino at the post-match press conference, was quick to defend the decision not to field a stronger side.
Perez said: "Always, we do the same. After the last game, we assessed our players and we decided the best team to play today.
"Obviously, it was a tough game against Hull, we played away as well, we came home late and some of our players were not fully recovered and we decided the best players to play today.
"For us, every single player is important and useful for us and it doesn't matter who is on the pitch.
"When we win, we win as a team. Today, we didn't get anything as a team as well.
"But I have nothing to say. There was a lack of freshness, that's obvious, but nothing more.
"When we decide to play with one or another player, it's because we trust them, they trust us. They support us, so there's nothing to complain about."
- Sports & Recreation