James Forrest's 114th-minute strike edged Celtic past Dunfermline after the Fifers threatened to end the champions' stranglehold on domestic silverware.
With a penalty shoot-out looming, the substitute worked space to steer a deflected effort into the far corner from 14 yards and send Celtic into the Betfred Cup quarter-finals with a 2-1 home win over the Ladbrokes Championship side.
Neil Lennon's side were looking for a lift following their European disappointment but they again started slowly and needed a wonder strike from Mikey Johnston to open the scoring 10 minutes into the second half.
But Celtic suffered a double blow when Kristoffer Ajer went off injured before Tom Beadling levelled in the 77th minute.
Dunfermline clung on during a one-sided extra-time but Forrest netted his fourth goal in as many games to make it 28 consecutive domestic cup wins for his team and keep alive the prospect of a fourth treble in a row.
The discontent among the Celtic support following their Champions League exit was evident in the much smaller and muted crowd and also a banner in the safe-standing section which accused the board of "downsizing" and "gambling 10 in a row".
Florian Kamberi shrugged off speculation regarding his future to inspire brittle Hibernian to a scarcely-merited 5-3 victory over Morton after extra-time.
The Swiss attacker, who has been linked with a £2 million switch back to his homeland with Basel, notched a brace to send the Hibees into the last eight despite another defensively dismal showing.
The hosts blew a comfortable 2-0 lead in the first half when an instinctive finish by Bob McHugh and a Lewis Stevenson own goal cancelled out strikes by Scott Allan and Josh Vela.
Hibs reclaimed their advantage after the break when Kamberi pounced on an error by Danny Rogers, only for Morton to level on the cusp of full-time when Steven Whittaker turned into his own net.
A controversial Kamberi effort in extra-time - scored amid furious claims for offside which ultimately saw Morton boss David Hopkin dismissed - and a late Christian Doidge goal secured Hibs' passage to the last-eight.
Livingston withstood a late comeback from League One outfit Forfar Athletic.
Lyndon Dykes gave the Ladbrokes Premiership side a first-half lead and Scott Pittman's second looked to give the Lions a comfortable cushion.
However, Dale Hilson gave Forfar hope in the 63rd minute and the visitors had to cling on for a 2-1 victory.
Kilmarnock squeezed past Hamilton after extra-time to ease the pressure on manager Angelo Alessio.
Three successive defeats, including the humiliating Europa League qualification exit to Welsh part-time outfit Connah's Quay, meant pressure was already building on Alessio.
But the Italian duly got the victory he needed as substitute Dom Thomas' strike in the second period of extra-time secured a crucial 1-0 win and saw Kilmarnock progress to the last eight of the competition.
Partick Thistle booked their place after a dramatic 3-2 extra-time win over Ross County.
The game looked destined for penalties until Steven Saunders rose to nod home a late winner.
James Penrice delivered a perfect corner allowing Saunders to leap above his marker and send a downward header beyond Ross Laidlaw to the delight of the home support.
