The 46-year-old Uruguayan guided his side past League One Carlisle with a 3-1 victory at the Stadium of Light on Sunday afternoon, but knew before half-time that the victors would face either Kidderminster or Peterborough in the next round.
Poyet was not unduly concerned by the situation, but wondered what might have happened had the draw paired the winners with one of the Barclays Premier League's giants.
He said: "I don't know if it's a good idea. It's an opinion - I don't want the FA to get mad with me because I'm not trying to go against anyone.
"I'm just saying if the draw was a top, top team from the Premier League at home, I am sure that Carlisle in the second half, maybe it would be a little bit different.
"If it's a replay, well there is no option, the draw has to be done. But with games still to be played, I don't know why they don't wait until tonight or tomorrow morning.
"If you are 1-1 at half-time and you get Arsenal at home and you are Carlisle, what a second half they are going to give you, that's for sure.
"If you are listening, please...it's just my opinion."
Poyet's comments came at the end of a better afternoon for the Black Cats as they secured a comfortable victory, their fifth in cup competitions so far this season, as opposed to just three in the league.
Adam Johnson's sumptuous 33rd-minute free-kick handed them the lead, but it lasted just nine minutes as boyhood Sunderland fan Matty Robson lashed home a shot off the underside of the crossbar with Wes Brown unable to clear on the line.
However, the home side needed just five minutes of the second half to regain their advantage when Carlisle skipper Sean O'Hanlon turned Johnson's firm cross past goalkeeper Greg Fleming, and substitute El-Hadji Ba's last-minute strike wrapped up the win.
Poyet, whose side has conceded five own goals to date this season, was relieved to see their luck change.
He said: "We were talking on the bench about it. I know Sean O'Hanlon quite well and he was very unfortunate. It went for us today."
The win was all the more pleasing for Poyet as he was forced to field midfielder Ki Sung-yueng as a central defender after seeing Modibo Diakite and Valentin Roberge join John O'Shea on the injured list.
He said with a smile: "We are going to become a cup team. It was a professional performance today. We needed to do the job.
"It's never easy. You need to cope with a few situations and a few shots, but in the end, you need to be better, so job done."
There were positives too for Carlisle counterpart Graham Kavanagh on his return to a club where he played under former manager Roy Keane, and his afternoon on Wearside reinforced his belief in the magic of the FA Cup.
He said: "It's huge. It's so exciting for a club like ourselves to come to a stadium as big as this with a fan-base as big as it is.
"For us then to get behind it because of the location and go and support the game and the team as they have done, and then always to feel that you have that chance of potentially going through and looking to see who you are going to draw...
"When we beat Brentford and the draw was made for Sunderland, the excitement around the city was huge.
"[Aston Villa boss] Paul Lambert has spoken about probably diminishing the FA Cup a little bit.
"I can totally understand where he is coming from as a manager because they need to stay in the Premier League for the financial gains, but the romance of the cup has always been, and will always be fantastic, and I hope we don't lose that."