The Easter Road outfit looked on course to rack up a comprehensive victory when a Stevie Mallan brace saw the hosts race into a two-goal lead within 12 minutes.
However, Jack Ross' men allowed the tempo to drop and the Highland League champions gradually found a foothold.
That was underlined in the second period when Brora, who had not kicked a ball since March, claimed one back through Dale Gillespie.
However, the gutsy visitors were unable to restore parity and ultimately saw Paul Hanlon make the game safe with an 88th minute header.
Mallan – one of eight changes to the Hibs side which defeated Hamilton last Saturday and making up for lost time on his first start of the season – wasted no time in staking his claim for more game time.
He fizzed a low drive beyond Joe Malin after dispossessing Colin Williamson, although the Brora keeper got a hand to it.
Mallan then doubled the hosts' lead when he met a Melker Hallberg Hibernian delivery and shot under the body of Malin.
Mallan and Jamie Gullan both went close in the remainder of the first half.
There was a note of concern for Ross after the break when Jamie Murphy abruptly pulled up clutching the back of his leg. He was immediately replaced by Martin Boyle.
Ross then had further cause for consternation when Gillespie lashed a fine free-kick beyond Barnes to halve arrears.
However, Hibs – while flat and ponderous – managed to make the game safe in the dying moments when Hanlon headed home from a fine Mallan corner-kick.