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 quarter-finals of the Betfred Cup 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.
However, six days on from their chastening 6-1 defeat against Rangers, their frailties were writ large once more.
Hibs broke the deadlock after 20 minutes when Glenn Middleton dashed down the left flank and found Allan, who fired a clinical low drive past debutant Rogers from the edge of the box.
The hosts swiftly doubled their advantage when Daryl Horgan danced to the byline and produced a perfect pin-point to allow Vela to blast home his first goal for the club since arriving in the summer.
McHugh halved arrears when he converted a dangerous Nicky Cadden delivery from close range, before parity was restored when Stevenson could only turn an Aidan Nesbitt cross into his own net.
Hibs restored their lead after the break when Rogers inexplicably dashed 40 yards from his line in a bid to sweep up a long pass from Vykintas Slivka. That proved an ill-fated decision by the on-loan Aberdeen keeper as the striker got to the ball first and slotted into an empty net.
However, Hopkin's side evidently do not know when they are beaten and dramatically levelled with just a minute left on the clock. Reghan Tumilty lashed a drive across the face of goal and the ball skewed into the roof of the net, with the final touch appearing to come off Whittaker.
Hibs were back in front in contentious fashion when Allan slipped a sumptuous through-ball to Kamberi, who showed admirable composure to slide a low shot past the on-rushing Rogers.
Hopkin, howling for an offside decision, was dismissed by referee Alan Muir for his complaints before insult was added to injury with Doidge's late strike.