The hosts scored the only goal and their goalkeeper saved a penalty to eject an Aberdeen team who took 73 minutes to register a shot on goal, but bombarded their hosts thereafter to no avail.
Instead, Hibs will travel to face Kilmarnock in the next round. And, in keeping with their peculiar past with a competition they have not won since 1902, it will mean each team they have faced this term – holders Hearts being the other – also provided William Hill Scottish Cup opposition last season.
Aberdeen were vanquished in a relatively perfunctory manner in the semi-finals in April but yesterday's meeting was much edgier.
Having established a lead just after the interval, Hibs faced a prolonged period of pressure as the Pittodrie side finally roused themselves.
Goalkeeper Ben Williams, who offered a belatedly outstanding display, was first called upon to save Russell Anderson's lash with 17 minutes remaining and then spent the rest of the match thwarting everything the visitors could throw at him.
Four very presentable chances fell to Aberdeen. Three of those, including a penalty, were spurned in the space of seven minutes by Scott Vernon. First, the substitute could only head straight at Williams after Niall McGinn popped the ball into his path; then he hesitated and was crowded out after Jonny Hayes located him with a cutback. Finally, having been presented with the opportunity to equalise from the spot with eight minutes remaining, the Englishman's effort was diverted on to a post by the goalkeeper, who gathered the rebound.
"Two weeks, two penalties," said Craig Brown, referencing Williams' spot-kick save from McGinn in the Clydesdale Bank Premier League game between the sides last weekend. "If we score them, we're fourth in the league and still in the cup."
Further saves from McGinn and Peter Pawlett further enhanced the goalkeeper's performance in an encounter that, ultimately, Aberdeen should have won.
Brown was adamant about that, just as he was about his own future. "Every game is a pressure game for Aberdeen," he said testily when pressed upon the matter.
The Aberdeen manager was equally evasive about why it took his side so long to test their hosts, who demonstrated more ambition for more of the contest. After an opening half so abject that it defies record, Hibs emerged determined to improve on a recent run of six matches in which they have scored just twice.
At the forefront was Leigh Griffiths. His dipping free-kick was pushed clear by Jamie Langfield and the goalkeeper was called into action again on the hour to beat away another fierce effort by the striker.
The custodian required the help of Hayes to prevent Clark Robertson's botched clearance skidding into his own goal but was helpless moments later when Hibs disturbed the tedium.
It was a goal entirely out of keeping with the drabness of the tie. There seemed little danger when Gary Deegan assumed possession on the halfway line, but the Irishman was allowed to meander forward to eventually unleash a stunning effort that dipped over Langfield and crashed in to the net behind him.
"I nearly lost me glasses in the celebrations," said Fenlon, who was bullish about the quarter-final draw.
The goal diffused the tension among the Hibs players. Paul Cairney attempted to score with a rabona only for Joe Shaughnessy to block, while Griffiths executed an outlandish scissors kick that Langfield gathered with relative ease.
It would be the last attacking impression the hosts made; after that the spotlight belonged to Williams.
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