City will head to the Pirelli Stadium for the second leg on January 23 but that tie is now little more than a formality after their demolition of the League One outfit on Wednesday.
Kevin de Bruyne opened the scoring with a header after just five minutes and Jesus doubled City's advantage with another headed effort on the half hour mark.
Jesus grabbed his second and City's third soon after and Oleksandr Zinchenko made it 4-0 before half time with a curling effort from range as the 22-year-old Ukrainian scored his first goal for the club.
Jesus completed his hat-trick on 57 minutes with a powerful header after good work from David Silva and Riyad Mahrez.
Ilkay Gundogan then picked out Jesus and he was denied by Brewers keeper Bradley Collins but substitute Phil Foden was on hand to pick up the pieces and give the hosts a 6-0 lead.
Jesus did grab a fourth soon after as he tapped home Leroy Sane's cross before goals from Kyle Walker and Mahrez completed the rout.
City, who are the defending champions, will face either Spurs or Chelsea in the final at Wembley on February 24.