Both sides started well in Staffordshire, but it was City who eventually took the lead on 38 minutes when James Justin cut in from the left wing and curled into the top corner for his second goal of the season.
The Foxes were clinical after the interval, doubling their lead via Marc Albrighton's neat finish before Ayoze Pérez and Harvey Barnes were both on target in the space of just three minutes as the visiting side booked their spot in the fourth round.
“I thought we were very good," Rodgers reflected to LCFC TV at full-time. “You see Stoke's results, Michael's [O'Neill] got them very well organised, they don't concede many goals. So, I thought our game today was at a really good level.
“We had to stay strong that first 15 minutes because they started well. We stayed together, showed a really good spirit and then once we got that first goal, that really opened up the game for us. It was a great goal by JJ [James Justin] and then we performed very well."