City boss Pep Guardiola named a fairly youthful side for the second leg given that the 9-0 success in the first game at the Etihad Stadium a fortnight ago meant the holders were already guaranteed their place in the Wembley final against either Tottenham or Chelsea on February 24.
The game was a lot closer than that first game and, although City dominated for large spells, the only goal came midway through the first half from Sergio Aguero.
Burton had a couple of chances to get on the scoresheet, although boss Nigel Clough will be proud of the way his players performed given that battering they suffered in the first leg.
Kyle McFadzean, Ben Turner, David Templeton, Kieran Wallace and William Miller were handed starts for Burton as part of six changes following their 2-2 draw with Doncaster in League One.
Manchester City boasted eight changes from Sunday's 3-0 win at Huddersfield with Sergio Aguero, Kevin de Bruyne, Riyad Mahrez and Phil Foden all handed starts, while England Under-19 winger Ian Carlo Poveda was given his debut and youngsters Arijanet Muric, Philippe Sandler and Eric Garcia also started.
City enjoyed some good early possession and went close in the 12th minute when Fabian Delph chipped the ball into the right side of the area for Mahrez and his effort from an acute angle whistled wide of the far post.
Burton managed an effort on target three minutes later when Liam Boyce met a Stephen Quinn free-kick from the right in front of goal and tried a cheeky flick which Muric got down to save.
Mahrez had another effort from an angle on 18 minutes which drifted wide of the far post, before City got their noses in front on the night in the 26th minute.
De Bruyne eluded a tackle and put the ball into the path of Mahrez, who then picked out Aguero in the middle of the area, and the prolific Argentinian hitman swept his effort into Bradley Collins’ net from 15 yards.
The hosts responded positively to going behind as Miller saw an effort from the left corner of the box drop not too far off the mark just before the half-hour mark.
However, City continued to control proceedings without really threatening to add to their tally until the 53rd minute when Danilo picked out Delph in the area, but he scuffed his initial volley and was then denied by Collins from the rebound.
Burton had a hopeful shout for a penalty waved away two minutes later when Templeton went to ground under a challenge from Danilo in the area, but referee Kevin Friend was confident in his decision enough not to consult with the Video Assistant Referee.
Aguero was guilty of an uncharacteristic miss on 61 minutes when he somehow pulled his shot wide of the goal from 10 yards after good work by Foden to tee him up.
Burton went close to an equaliser on the night in the 70th minute when a slip by Sandler played in Boyce, who then saw his effort in the box well saved by Muric. The rebound dropped kindly to Miller but his close-range strike was blocked on the line by Garcia.
City had a couple of chances to add to the scoreline late on but Collins did well to block substitute Gabriel Jesus’ close-range effort, before the Brewers’ custodian tipped over a 25-yard free-kick from Danilo in stoppage time.