The Championship promotion contenders have been drawn at home to the winner of the second-round replay between League One outfit Peterborough or National League side Woking.
“The good thing is we’re at home,” Bruce said.
“It’s a bit of a change for me as I usually pull out Tottenham, Arsenal or Liverpool.
“It’s an outstanding chance for us to progress.”
Before they begin their FA Cup campaign on January 6, Villa have a string of Championship fixtures to navigate.
Bruce believes the hectic Christmas schedule is crucial to their promotion hopes, and he said: “If I ever need reminding about the size of this club, I’d only have to think back to Boxing Day last year when we played Burton Albion and there was 42,500 here.
“The fans enjoy Christmas and coming to watch us at home.
“It’s a big, important time coming up and they can more than play their part.”
A fine run of results has lifted Villa to fifth in the Championship table but they are seven points behind second-placed Cardiff City.
Millwall are the visitors to Villa Park tomorrow (3pm) and Bruce is expecting a tough clash.
“It’ll be a big one,” he said. “Neil Harris has done a terrific job there.
“You know what you’re going to get – there’s a demand that they play a certain way.
“Physically we’ll have to be ready and we know they’ll enjoy the challenge of coming to Villa Park themselves.
“As always, it’ll be a tough game.”