The Foxes will head to Kent for the Saturday lunchtime kick-off still stinging from last season's defeat to local rivals Nottingham Forest, who were then a Championship club.
This ought to be an easier assignment, given the Gills prop up the entire Football League, but Rodgers is not taking the trip lightly.
"We want to go as far as we can. Like I've always said, we want to win every cup competition we're in," he said.
"We arrive into the game against a team in Gillingham that hasn't done as well as they would have liked in the league but certainly in the cup competitions, they have made it very difficult for Premier League teams.
"You go away in this competition, you have to be strong, you have to be aggressive and you have to be professional in your job.
"These are dangerous games if you're not at maximum concentration and maximum commitment, you can come unstuck.
"We've always tried to present that to players. There are no easy games. If you want to make changes, the players coming in have to be ready and prepared for that."
Leicester were humbled in the East Midlands derby in last season's fourth round where their defence of the trophy they won in the 2020/21 season ended in the fourth round.
Rodgers says that game at the City Ground will be a reference point to him in the build-up to the game at the Priestfield.
"No, you always have to reinforce and remind," he said.
"You might have new players, you might have young players playing, so it's always an opportunity for a team at a different level to beat the Premier League team.
"Like we did in the last round of the League Cup, you've got to go and do a professional job, and you have to have that intensity, that work, and that mentality. Hopefully then your quality will come through."