Morton appeared to be heading through to the third round courtesy of strikes from Iain Russell and Stevie Masterton.
However, Joe Keenan finally dug out a goal to spark a Hibs fightback and Dean Shiels' 85th-minute equaliser took the match into extra-time.
Pinau then needed only 36 seconds to put Mixu Paatelainen's men into the lead for the first time in the match but an equaliser from Russell from the spot and Ryan Harding's winner sealed it for Davie Irons' men.
Colin Nish almost opened the scoring in the first minute and at the other end goalkeeper Andy McNeil had to dive low to his left to keep a shot from Iain Russell out in a bright opening spell.
Steven Fletcher should have done better than hit John Rankin's pass straight at Kevin Cuthbert and Nish then struck the bar from a corner as Hibs sought the breakthrough.
Fletcher was again denied by Cuthbert in the 26th minute following a one-two with Nish and three minutes later Morton went up the other end and scored.
Russell won the chase with McNeil for McAlister's through ball and from the tightest of angles speared his shot past Ian Murray's despairing dive on the line.
Only the crossbar denied Morton a second as Stevie Masterton's inswinging corner was deflected in a crowded goalmouth.
Fletcher spurned a second one-on-one with Cuthbert, again striking the keeper with his shot when he should have eased nerves with the equaliser.
With teenager David Grof handed his first-ever competitive outing as a replacement for the injured McNeil, things went from bad to worse for the home side.
Hibs manager Mixu Paatelainen introduced Merouane Zemmama and he was denied twice in quick succession.
There was nothing home keeper Grof could do at the other end in the 66th minute, however, as Morton took a two-goal lead.
Lewis Stevenson fouled Russell 30 yards out and Masterton curled a magnificent effort into the top corner.
The match appeared to be slipping away from Hibs when Keenan drove high into the roof of the net with 11 minutes remaining.
Shiels then sent the match into extra-time as he weaved his way into the box and beat the defiant Cuthbert with a low shot into the bottom corner.
When substitute Steven Pinau netted his first goal for the club, it appeared Hibs would be going through until Morton's magnificent late fight back.