Burton, managed by Nigel Clough, the son of former Nottingham Forest manager Brian, must endure another 90 minutes at their own ground on Jan. 23.
City are set to face the winners of the other semi-final between Chelsea and Tottenham Hotspur, which Spurs lead 1-0 from the first-leg.
City manager Pep Guardiola fielded a strong starting line-up and his side quickly established their dominance with Kevin De Bruyne steering in a header from an excellent lofted ball from David Silva to open the scoring in the fifth minute.
It was 4-0 by halftime after two goals from Jesus and a long-range lob from Oleksandr Zinchenko and City showed no mercy after the break. Jesus added two more and substitute Phil Foden, Kyle Walker and Riyad Mahrez also found the net.
The result was the largest ever margin of defeat in the semi-final of the competition.