Martin Scott scored twice, starting the rout after 11 minutes, with Akpo Sodje adding a second after 38 minutes as Hibs, struggling in the SPL, ended hopes of an upset in the opening half.
Garry O'Connor headed in his fourth of the season six minutes into the second half, Scott added his second after 58 minutes and Ivan Sproule struck his first since returning to Edinburgh after 87 minutes as Irn-Bru Third Division Berwick returned south down the A1 empty handed.
Hibs' attention now returns to the SPL and Sunday's Edinburgh derby with Hearts at Tynecastle.
Pressure has been building on Calderwood, who now has a record of 10 wins in more than 10 months as Hibs boss and has been linked with a move away from Leith, and a side seventh in the bottom tier were welcome visitors.
The evening could have been so different for part-time Berwick, who beat Partick Thistle in the first round, had Fraser McLaren not dragged an effort wide after two minutes following a run in behind Hibs' makeshift right-back David Wotherspoon.
Sodje, one of five changes to the starting line-up which began the 2-1 loss to St Mirren last weekend, spurned a glorious chance when played through by O'Connor, shooting straight at Jamie Barclay.
But two minutes later the Rangers goalkeeper was beaten by Scott's 25-yard shot from the right channel which was driven into the far corner.
Lee Currie played Damon Gray through on goal as the visitors attempted to respond, but the striker's weak effort was saved by Mark Brown.
Lee Currie then slid into his own area to intercept Sproule's cross, the post sparing him from an embarrassing own goal against his former team.
Hibs went two up soon after as Sproule burst down the right, fed O'Connor who played an angled ball into the box which Sodje collected and tucked in.
The hosts netted a third after 51 minutes, Sproule given freedom to cross from the right and find O'Connor, who had time to pick his spot.
Moments after Hibs left-back Scott Smith, making his debut as a half-time substitute for Calum Booth, had an effort tipped onto the bar, Scott netted his second.
The midfielder pulled down a long cross from Smith, made space for himself and fired in from inside the right edge of the area.
Smith was proving useful on the left and Sodje should have converted one cross after 63 minutes before Rangers replacement Darren Gribben shot wide at the other end and Paul Currie had an effort saved by Brown.
Hibs' defence looked suspect and Paul Currie lofted an effort over with the goal gaping before Sodje missed at the other end after intercepting Stephen Thompson's attempted pass back.
Barclay saved from Scott and then Sean O'Hanlon's header before Sproule netted a fifth after pulling down a long ball forward to complete the win.