Motherwell scored twice in the final eight minutes to seal a 2-2 draw but Hibs triumphed 4-2 on spot-kicks to reach the last four in a fourth consecutive cup campaign.
Hibernian looked to be heading to Hampden without much trouble when Jackson Irvine added to Christian Doidge's opener with 10 minutes left amid offside suspicions.
But Ricki Lamie and substitute Tony Watt scored to send the game into extra-time.
After an intriguing extra 30 minutes, there was more drama. Mark O'Hara blazed over and Steven Lawless hit a post to put Hibs in the driving seat in the shoot-out before Motherwell goalkeeper Liam Kelly saved from Kevin Nisbet.
Ryan Porteous kept his nerve to send Hibs into the semi-finals in a fortnight's time.
Motherwell continued with the same team and 3-5-2 formation which earned victory at Hamilton and they only came alive after making some attacking changes in the final quarter of normal time.
Hibernian's attacking triumvirate of Martin Boyle, Nisbet and Doidge had netted all 13 of their team's goals in the previous six matches and they looked lively in the opening stages.
Boyle and Nisbet put in a couple of dangerous crosses each and the latter fired wide from 22 yards.
Doidge and Irvine headed wide late in the half and the visitors stepped up a gear after the interval before taking the lead in the 52nd minute.
Doidge ran on to Paul McGinn's inviting cross and got beyond Tyler Magloire to head past Kelly from 12 yards.
Doidge soon had a glorious chance when Irvine headed down Boyle's cross but the striker stabbed over from close range.
Devante Cole created a great chance to level when he skipped past a defender but Macey diverted the shot wide.
Graham Alexander brought on striker Watt, who has been missing recently with an ongoing heel problem, and attacking midfielder Lawless, and brought off defender Magloire and anchor man Robbie Crawford.
But Hibs went two ahead. A deep cross from the left ultimately fell for Boyle, who cut it back for Nisbet after being confronted by Kelly. The striker scuffed a shot which Irvine helped home from inside the six-yard box amid offside appeals.
Motherwell replied inside two minutes. A corner fell for Lamie on the far side of the box and he cushioned a volley over a crowded goalmouth and into the far corner of the net.
The visitors levelled with two minutes left when Watt headed home Stephen O'Donnell's looping cross.
Joe Newell had a stoppage-time chance to win it for Hibs but Kelly saved to set up another half hour of action.
Motherwell retained the attack-minded line-up which got them back into the game and it made for an open extra-time.
O'Donnell nearly set up another goal but Lawless and Cole just could not reach his inviting cross and Kelly brilliantly denied Boyle with three minutes left after a mix-up between Declan Gallagher and Nathan McGinley had gifted him a chance.
Tempers frayed in the closing moments after McGinley rugby-tackled Hibs substitute Melker Hallberg, who appeared to kick the full-back.
A shoving match ensued and resulted in a yellow card for the Motherwell full-back and a second yellow card for Alex Gogic, who had to watch his team triumph from the dressing room.