Jason Cummings, Lewis Stevenson, Sam Stanton and Liam Fontaine decided the quarter-final for the Championship high-fliers, who were runners-up in the competition in 2012 and 2013.
Cummings and Stevenson were on target twice inside two first-half minutes with left-footed strikes as the hosts swept Berwick, who were aiming to reach the last four for the first time, aside.
The returning Cummings slotted the opener in the 27th minute after running on to a pass from Stevenson who, soon afterwards, exchanged passes with Dominique Malonga before running on to score.
Substitute Stanton added a third goal midway through the second half, providing a simple finish after Dominique Malonga’s effort was saved by visiting goalkeeper Billy Bald.
Liam Fontaine made certainty of Hibs’ safe progress into the last four when scoring a fourth eight minutes from time, with the defender heading past Bald from Stanton’s corner.