The Lions were drawn to face Paul Lambert's Premier League side in the fourth round after seeing off Preston on Saturday.
Liam Feeney's first-half goal was enough to secure a plum home tie at the end of January.
Millwall may be in the thick of the play-off race in the Championship but Jackett is determined to make the most of the FA Cup as well.
"I'm very pleased to have drawn a Premier League club of the tradition and status of Aston Villa, particularly at home," Jackett told the club's official website.
"It gives us all something to look forward to at the end of the month and between now and then we can focus on some important Championship fixtures."
The cup win did come at a cost, though, with goalscorer Feeney likely to be out for a couple of weeks with a hamstring injury.