The home side found themselves 4-2 up at half-time despite being outplayed by the students for part of the opening period.
Calvin Bisset fired in the opener for Fernie from a Josh Neill assist, before the students hit back to level.
Craig Perry regained their lead and it was soon doubled as Bisset netted his second of the afternoon. Grant Scott got Fernieside’s fourth but Edinburgh Uni pulled another back ahead of the interval.
The first goal in the second period was going to be crucial, and it was the hosts that got it, courtesy of Cammy Forbes’ free-kick.
Forbes then got his second assist of the day when Rab Gillespie headed in his cross, before Jamie Blackie fired in their seventh. Forbes was shown a red card late on but they were well on their way to the last four by that stage.
Meanwhile, Edinburgh Star are through to the semi-finals of the Saturday’s RJM Sports East of Scotland Cup after a 3-1 defeat of Steins Thistle.
Star led 2-1 at the interval after Jak Townsley’s effort from five yards and a James Guy deflected cross cum shot, either side of a Steins equaliser.
Keeper Kerr Robertson saved a penalty, before substitute Ross Forrest headed in Coleman’s corner to make sure of the win.
Corstorphine Dynamo beat Gorgie Hearts 3-1 in Sunday’s Premier Division to solidify their grip on third place in the table.
Striker Stuart Donaldson found the far corner of the net with a fierce strike across goal after being released by Sean Forsyth.
Gorgie levelled from the penalty spot midway through the first half and the score remained the same until six minutes into the second period as Donaldson struck again after Sean Morrison had put him through.
Morrison then added a goal of his own after Forsyth picked him out. Gorgie pushed to get back into the match but found Dynamo goalkeeper Stuart Scott in top form.