It was 0-0 at half time, but Rangers teenager Barrie Mackay scored with his first touch of the game on the hour mark, just as Berwick might have been thinking about stealing the game themselves.
Ian Little’s men had dampened much of the hosts’ attacking intent up until then.
For the first 20 minutes the two teams sounded each other out, and after Andrew Little’s weak volley, Rangers appeared to settle for long periods of sustained, but sterile, possession.
They were spurred into action by the jeers of the home support, and while they created some good chances, all went begging.
Most of the home side’s best work came from David Templeton, who provided the chances that Little and MacLeod spurned.
The visitors had their hearts in their mouths on one occasion, but with keeper Paul Grant on his back and the ball trickling towards goal, Dean Hoskins stepped in to make a goal-line a clearance.
As the half wore on, Berwick grew in confidence, with Darren Lavery pushing at the offside trap and looking less isolated up front.
While much of Berwick’s best play was on the break, they will draw confidence from the ball-stealing work of Devon Jacobs, in particular, who appears keen to turn defence into attack at every opportunity.
Home manager Ally McCoist was looking for something similar from his left back, Emilson Cribari. But the delicate nature of cup football was made painfully obvious, as Lavery was released immediately into the space the defender vacated.
Both sides came out for the second half looking to make their attacks count, and as both defences got stretched, chances became more clear cut.
For Rangers, Robbie Crawford got far too much space and time on the right wing,and he would punish Berwick.
With one of Crawford’s runs repelled, the Berwick defence was still stretched when Barrie MacKay, having only come on as a sub a minute before, flicked the ball nonchalantly past Grant.
The lead was doubled a minute later. Crawford jinked in from the right wing and stood the ball up for Little, who nodded in without having to break his stride or even jump.
Berwick continued to make a game of it despite the double setback, but with Rangers finally playing with confidence, the result was never in doubt.
Rangers: Gallacher; Faure, McCulloch, Cribari, Wallace; Crawford, Black, Macleod, Templeton (McAusland 77); Shiels (McKay 59), Little (Aird 72). Subs not used: Kelly, Gasparotto.
Scorers: McKay (60), Little (62)
Berwick Rangers: Grant; Jacobs, Dunlop, Tulloch, Hoskins; Gray (Morris 56), Notman, Janczyk (Dalziel 70), Currie, O’Brien (Gelty 70); Lavery. Subs not used: Bald, Fairbairn.
Referee: Euan Norris