James Milner broke the deadlock at Stadium MK in the 41st minute when his shot from the left corner of the area squirmed from the grasp of goalkeeper Stuart Moore and crossed the line.
The Reds were in control for much of the contest, though the home team threatened on several occasions, coming closest when Jordan Bowery prodded against the base of the post in the second half.
However, Ki-Jana Hoever’s powerful header from a Milner cross on 68 minutes put the tie beyond the reach of the League One club as the 17-year-old registered his first senior goal to send Jürgen Klopp’s side into the next round of the competition.