Berwick failed to deliver a shock to match their famous victory over Glasgow Rangers back in 1967 as they went out at Shielfield Park.
Neil Lennon's men have suffered their fair share of upsets in the competition in the past but goals either side of the interval from Daniel Majstorovic and Scott Brown saw them safely into Tuesday afternoon's fifth-round draw.
Berwick wrote their names into the history books 44 years ago this month when they pulled off arguably the biggest shock in the history of Scottish football when they dumped Rangers out of the competition.
More than four decades on, Jimmy Crease had the opportunity to match that stunning achievement when the other half of the Old Firm came calling, although one look at the Celtic team-sheet would have left the Irn-Bru Third Division side under no illusions about the size of task they faced.
The Hoops made just two changes from the side who triumphed in last week's Old Firm derby, with Freddie Ljungberg recovering from a virus to make his debut in place of Emilio Izaguirre.
Lukasz Zaluska took over from Fraser Forster between the sticks, while captain Brown - back from suspension - had to settle for a place on the bench.
Battling Berwick looked up for the tie from the outset and almost caused problems early on when Zaluska was forced to race yards out of his box to clear the danger before Darren Gribben could connect with a dangerous long ball.
Gribben then had the ball in the back of the net after lobbing the Celtic goalkeeper but, unfortunately for the striker, the offside flag was raised well before the shot even left his boot.
At the other end, Paddy McCourt saw his header flash inches past the upright as Celtic began to impose themselves on the game.
The breakthrough came after 17 minutes when Andy McLean failed to deal with a Charlie Mulgrew cross from the right and Majstorovic was on hand to rifle into the roof of the net from just a few yards out.
The Wee Rangers had the chance to level the scoring when Thomas Rogne conceded a free-kick 25 yards out but Gribben wasted the opportunity when he blasted well over, failing even to test Zaluska.
Berwick goalkeeper Mark Peat had passed a late fitness test ahead of the game and he pulled off an impressive save to prevent a point-blank effort from James Forrest from finding the back of the net as the Glasgow giants were forced to settle for a single goal advantage heading into the break.
The Hoops also suffered a setback when Georgios Samaras - who netted both goals against Rangers - limped out of the action and was replaced by Anthony Stokes before half-time.
The home side emerged for the second half eager to haul themselves back into the tie and Zaluska was called into action to save at the feet of David Greenhill shortly after the restart.
Celtic made their second change with just over an hour gone when Ljungberg was withdrawn for Brown.
It was McCourt who could have doubled the advantage moments later when he met a cross from the right from Forrest at the back post only to see his header comfortably gathered by the Berwick goalkeeper.
The Irishman was then off target, this time connecting with Mulgrew's corner, before bulleting a header across goal and wide.
Any hopes of a dramatic comeback where dashed with eight minutes to go when Brown raced on to a Beram Kayal ball before sending an angled drive past Peat and into the bottom corner.
Stokes also had the ball in the back of the net moments later but was denied adding his name to the scoresheet by the assistant referee's offside flag, with Celtic by then ensured of their progression in the competition.