The tie was already over after City's remarkable 9-0 first-leg win but Pep Guardiola still named several first-team stars in a weakened line-up.
Sergio Aguero scored the only goal of the night from Riyad Mahrez's pass after good play from Kevin de Bruyne.
Burton did have chances with Will Miller's shot cleared off the line.
EFL Cup holders City will face either Chelsea or Tottenham in the Wembley final on Sunday, 24 February.
Spurs lead 1-0 going into Thursday's semi-final second leg against the Blues at Stamford Bridge.
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